Optimal Casorati Inequalities for Pointwise SlantRiemannian Maps

Abstract

We establish optimal Casorati inequalities for pointwise slant Riemannian maps from Riemannian manifolds into generalized complex space forms and generalized Sasakian space forms. The resulting bounds relate the normalized horizontal scalar curvature to the normalized Casorati curvatures and the ambient constants, with an explicit dependence on the slant angle $\theta$. For a map of rank $r\ge 3$ into a generalized complex space form $M(c_1,c_2)$, the bound reads ${\rho }^{\mathcal{H}}\le {\delta }_C^{\mathcal{H}}(r-1)+c_1+{\displaystyle \frac{3c_2}{r-1}}{cos}^2\theta$. We also characterize totally geodesic pointwise slant Riemannian maps via simultaneous equality in both inequalities. As applications, sharp non-existence obstructions are obtained for totally geodesic pointwise slant proper Riemannian maps from flat Euclidean space into standard Sasakian and Kenmotsu space forms.

1. Introduction

The study of curvature invariants has been central to differential geometry since Gauss’s foundational work on surface curvature. While Gaussian curvature provides fundamental information about surfaces, it may vanish even on curved surfaces, limiting its effectiveness. To address this, Casorati [1] introduced in 1890 the concept of Casorati curvature, an extrinsic invariant that vanishes only at planar points, making it particularly useful for visualization and shape analysis [2,3].

The modern development of Casorati inequalities began with Decu et al. [2], who established optimal inequalities relating normalized Casorati curvatures to scalar curvatures for submanifolds in real space forms, inspired by Chen’s theory of $\delta$-invariants [4,5]. Since then, numerous geometers obtained Casorati inequalities for submanifolds in a wide variety of ambient spaces: Aquib et al. [6,7] for Lagrangian and bi-slant submanifolds in generalized complex space forms, Lee et al. [8,9,10] for Legendrian submanifolds in Sasakian and Kenmotsu space forms, Lone [11,12,13,14] for slant submanifolds in generalized complex and Sasakian space forms, Vilcu [15] for Lagrangian submanifolds in complex space forms, Zhang et al. [16,17] for submanifolds in real and complex space forms, and Siddiqui [18] for bi-slant submanifolds in generalized Sasakian space forms. For a comprehensive survey see Chen [5].

Fischer [19] introduced Riemannian maps, which simultaneously generalize isometric immersions and Riemannian submersions [20,21,22], and satisfy the eikonal equation, which is a bridge between geometric and physical optics. Sahin [23,24,25,26] developed a comprehensive theory of Riemannian maps in Hermitian geometry, building on Vanhecke’s study of J-invariant curvature tensors [27]. Casorati inequalities for Riemannian maps and submersions were first obtained by Lee et al. [28] for real and complex space forms, then extended by Polat et al. [29] to Sasakian space forms and by Zaidi–Shanker [30]. Fatima et al. [31] treated invariant maps to nearly Kähler manifolds. Recently, Singh et al. [32] established the general Casorati inequality for Riemannian maps and Riemannian submersions and then specialized it to the invariant and anti-invariant cases; the corresponding result for Riemannian submersions, treating the horizontal and vertical Casorati curvatures separately, was subsequently developed by Singh [33]. Notably, in all previous studies, the structure tensor was constrained to act either invariantly ($\theta =0$) or anti-invariantly ($\theta =\pi /2$) on the relevant distribution; the intermediate case, where the angle varies pointwise, remained open. A pointwise slant Riemannian map allows the angle $\theta$ between the structure and the range space to vary pointwise on $M_1$, with $\theta \equiv 0$ and $\theta \equiv \pi /2$ as the classical extremes.

The notion of slant submanifolds was introduced by Chen [34] as a natural generalization of invariant and anti-invariant submanifolds, which was extended to contact geometry [35,36]. Etayo [37] introduced pointwise slant submanifolds, where the angle varies point-to-point.

Pointwise slant Riemannian maps to almost Hermitian manifolds were introduced and studied by Gunduzalp and Akyol [38]. Pointwise slant Riemannian maps from almost contact metric manifolds have recently been treated by Demir–Yildirim [39]. For pointwise slant submersions Kumar-Prasad [40] and Prasad-Kumar [41] developed the theory from almost contact manifolds. These structures unify invariant, anti-invariant, and all intermediate cases with varying slant angle.

The corresponding theory for Riemannian submersions, in which $F_{\ast }$ is surjective and the extrinsic geometry splits between O’Neill’s tensors T and A on the vertical and horizontal distributions, is developed in the companion paper [42]. Although Riemannian maps generalize submersions, the submersion inequalities are not a special case of the present ones: when ${(Ran{F}_{\ast })}^{\perp }=0$ the single second fundamental form $\nabla F_{\ast }$ vanishes and the curvature data instead distributes across two tensors on orthogonal distributions, which must be optimized separately. Here we treat the single-tensor case, where total geodesicity is characterized by simultaneous equality in both Casorati bounds.

Despite extensive development of both Casorati inequalities for maps and pointwise slant Riemannian map theory, Casorati inequalities for pointwise slant Riemannian maps have not been established. This gap is the subject of the present paper. The key challenge is that when $\theta$ is variable, computing the range-space scalar curvature ${\rho }^{\mathcal{R}}$ requires controlling the tangential projection operator P through the identity ${\parallel P\parallel }^2=r{cos}^2\theta$, which we establish in Proposition 2. Applying this identity in the Gauss equation yields Casorati inequalities with an explicit dependence on the slant angle $\theta$.

The paper is organized as follows. Section 2 recalls Riemannian maps, space forms, and the algebraic optimization lemma used in the proofs. Section 3 defines the pointwise slant Riemannian map to almost Hermitian as well as almost contact manifolds. Section 4 contains our main results, focusing on the cases where the base manifold is a generalized complex space form or a generalized Sasakian space form, including explicit corollaries and limiting propositions. Section 5 presents examples illustrating the sharpness of the inequalities. Section 6 concludes with a comparison to the literature and open problems.

2. Preliminaries

In this section, we recall the fundamental concepts needed for our main theorems. Throughout, all manifolds are assumed to be smooth and connected.

Let $F:(M_1^{m_1},g_1)\to (M_2^{m_2},g_2)$ be a smooth map between Riemannian manifolds. For a point $p\in M_1$, denote by $F_{\ast p}:T_p{M}_1\to T_{F\left(p\right)}{M}_2$ the derivative map. Let $ker{F}_{\ast p}$ be the kernel of $F_{\ast p}$ and ${(ker{F}_{\ast p})}^{\perp }$ its orthogonal complement in $T_p{M}_1$. Then we have the orthogonal decomposition

$$T_p{M}_1=ker{F}_{\ast p}\oplus {(ker{F}_{\ast p})}^{\perp }.$$

(1)

Similarly, at $F\left(p\right)\in M_2$, let $Ran{F}_{\ast p}$ be the range of $F_{\ast p}$ and ${(Ran{F}_{\ast p})}^{\perp }$ its orthogonal complement in $T_{F\left(p\right)}{M}_2$, giving

$$T_{F\left(p\right)}{M}_2=Ran{F}_{\ast p}\oplus {(Ran{F}_{\ast p})}^{\perp }.$$

(2)

Definition 1

([19]). F is called a Riemannian map at $p\in M_1$ if the linear map $F_{\ast p}:{(ker{F}_{\ast p})}^{\perp }\to Ran{F}_{\ast p}$ is a linear isometry; that is,

$$g_1(X,Y)=g_2(F_{\ast }X,F_{\ast }Y)\mathit{for}\mathit{all}X,Y\in \Gamma ({(ker{F}_{\ast })}^{\perp }).$$

(3)

If (3) holds at every point $p\in M_1$, then F is called a Riemannian map.

When ${(Ran{F}_{\ast })}^{\perp }=\left\{0\right\}$, the map F is a Riemannian submersion; when $ker{F}_{\ast }=\left\{0\right\}$, it is an isometric immersion. Thus Riemannian maps simultaneously generalize isometric immersions and Riemannian submersions.

Let ${\nabla }^{M_1}$ and ${\nabla }^{M_2}$ denote the Levi-Civita connections on $M_1$ and $M_2$, respectively. The second fundamental form of F is defined by [43]

$$(\nabla F_{\ast })(X,Y)={\nabla }_{F_{\ast }X}^{M_2}{F}_{\ast }Y-F_{\ast }\left({\nabla }_X^{M_1}Y\right),X,Y\in \Gamma \left(T{M}_1\right).$$

(4)

Proposition 1

([23]). For a Riemannian map F, the second fundamental form satisfies

$$(\nabla F_{\ast })(X,Y)\in \Gamma ({(Ran{F}_{\ast })}^{\perp })\mathit{for}\mathit{all}X,Y\in \Gamma ({(ker{F}_{\ast })}^{\perp }).$$

(5)

For $V\in \Gamma ({(Ran{F}_{\ast })}^{\perp })$, the shape operator ${\mathcal{S}}_V$ is defined by

$${\nabla }_{F_{\ast }X}^{M_2}V=-{\mathcal{S}}_V{F}_{\ast }X+{\nabla }_X^{F^{\perp }}V,$$

(6)

where ${\nabla }^{F^{\perp }}$ is the connection in the normal bundle. The shape operator relates to the second fundamental form by

$$g_2((\nabla F_{\ast })(X,Y),V)=g_2({\mathcal{S}}_V{F}_{\ast }X,F_{\ast }Y).$$

(7)

Theorem 1

([26]). For a Riemannian map $F:M_1\to M_2$ and for $X,Y,Z,W\in \Gamma \left({(ker{F}_{\ast })}^{\perp }\right)$, the Gauss equation is

$$\begin{array}{c}{g}_2(R^{M_2}(F_{\ast }X,F_{\ast }Y)F_{\ast }Z,F_{\ast }W)=g_1(R^{M_1}(X,Y)Z,W)\\ +g_2((\nabla F_{\ast })(X,Z),(\nabla F_{\ast })(Y,W))-g_2((\nabla F_{\ast })(X,W),(\nabla F_{\ast })(Y,Z)),\end{array}$$

(8)

where $R^{M_1}$ and $R^{M_2}$ denote the Riemannian curvature tensors of $M_1$ and $M_2$, respectively.

We shall use the following elementary fact, if $\{e_1,\dots ,e_r\}$ is an orthonormal basis of an inner product space V and $T:V\to V$ is a linear operator, then

$${\parallel T\parallel }^2:={\displaystyle \sum _{i,j=1}^r}g{(T{e}_i,e_j)}^2={\displaystyle \sum _{j=1}^r}{\parallel T{e}_j\parallel }^2.$$

This is Parseval’s identity for the operator norm. We now recall the ambient spaces used throughout the paper.

Definition 2

([44,45]). An almost Hermitian manifold $(M,g,J)$ is called a generalized complex space form, denoted $M(c_1,c_2)$, if its curvature tensor satisfies

$$\begin{array}{ccc}\hfill R(X,Y)Z& =& c_1\{g(Y,Z)X-g(X,Z)Y\}\hfill \\ & & +c_2\{g(X,JZ)JY-g(Y,JZ)JX+2g(X,JY)JZ\}\hfill \end{array}$$

(9)

for smooth functions $c_1,c_2$ on M.

Important special cases include

(i)

Real space forms: $c_1=c$, $c_2=0$ (constant sectional curvature c).

(ii)

Complex space forms: $c_1=c_2=c/4$ (Kähler, constant holomorphic sectional curvature c).

(iii)

Real Kähler space forms: $c_1=(c+3\alpha )/4$, $c_2=(c-\alpha )/4$.

Definition 3

([35,46]). An almost contact metric manifold $(M,\varphi ,\xi ,\eta ,g)$ is called a generalized Sasakian space form, denoted $M(c_1,c_2,c_3)$, if

$$\begin{array}{ccc}\hfill R(X,Y)Z& =& c_1\{g(Y,Z)X-g(X,Z)Y\}\hfill \\ & & +c_2\{g(X,\varphi Z)\varphi Y-g(Y,\varphi Z)\varphi X+2g(X,\varphi Y)\varphi Z\}\hfill \\ & & +c_3\{\eta \left(X\right)\eta \left(Z\right)Y-\eta \left(Y\right)\eta \left(Z\right)X+g(X,Z)\eta \left(Y\right)\xi -g(Y,Z)\eta \left(X\right)\xi \}\hfill \end{array}$$

(10)

for smooth functions $c_1,c_2,c_3$ on M.

Special cases include

(i)

Sasakian space forms: $c_1=(c+3)/4$, $c_2=c_3=(c-1)/4$.

(ii)

Kenmotsu space forms: $c_1=(c-3)/4$, $c_2=c_3=(c+1)/4$.

(iii)

Cosymplectic space forms: $c_1=c_2=c_3=c/4$.

(iv)

Almost $C\left(\alpha \right)$ space forms: $c_1=(c+3{\alpha }^2)/4$, $c_2=c_3=(c-{\alpha }^2)/4$.

We shall use the following algebraic optimization lemma due to Tripathi [47]. It identifies the constrained minimum of a specific quadratic form on a hyperplane in ${\mathbb{R}}^r$.

Lemma 1

([47]). Let $\Lambda =\{(z_1,\dots ,z_r)\in {\mathbb{R}}^r:z_1+\cdots +z_r=k\}$ be a hyperplane in ${\mathbb{R}}^r$, and let $f:{\mathbb{R}}^r\to \mathbb{R}$ be the quadratic form

$$f(z_1,\dots ,z_r)={\lambda }_1{\displaystyle \sum _{i=1}^{r-1}}{z}_i^2+{\lambda }_2{z}_r^2-2\sum _{1\le i<j\le r}{z}_i{z}_j,$$

(11)

where ${\lambda }_1,{\lambda }_2>0$ satisfy ${\lambda }_2=(r-1)/({\lambda }_1-r+2)$. Then the constrained minimum of f on Λ is achieved at

$$z_1=\cdots =z_{r-1}={\displaystyle \frac{k}{{\lambda }_1+1}},z_r={\displaystyle \frac{k({\lambda }_1-r+2)}{{\lambda }_1+1}},$$

(12)

and at this critical point $f(z_1,\dots ,z_r)=0$.

Recall that a submanifold is called quasi-umbilical if its shape operator has at most two distinct eigenvalues, one of multiplicity $n-1$ and one of multiplicity 1 [2]. For a distribution in a Riemannian map, this condition translates to

$$h_{11}^{\alpha }=\cdots =h_{r-1,r-1}^{\alpha }={\displaystyle \frac{1}{2}}{h}_{rr}^{\alpha },h_{ij}^{\alpha }=0(i\ne j),$$

for all $\alpha$.

Remark 1.

The condition ${\lambda }_2=(r-1)/({\lambda }_1-r+2)$ requires ${\lambda }_1>r-2$ for ${\lambda }_2>0$. In our applications, Lemma 1 is used with ${\lambda }_1=r$ and ${\lambda }_2=(r-1)/2$ (see the proofs of Theorems 2 and 3). Since $r\ge 3$, we have ${\lambda }_1=r>r-2$, so the positivity condition is automatically satisfied. With ${\lambda }_1=r$, the relation gives ${\lambda }_2=(r-1)/2$ directly. The critical point (12) then becomes

$$z_1=\cdots =z_{r-1}={\displaystyle \frac{k}{r+1}},z_r={\displaystyle \frac{2k}{r+1}},$$

which corresponds to the quasi-umbilical ratio $h_{11}^{\alpha }=\cdots =h_{r-1,r-1}^{\alpha }={\displaystyle \frac{1}{2}}{h}_{rr}^{\alpha }$ in the equality cases of Theorems 2 and 3. This is precisely the quasi-umbilical condition for the horizontal distribution: the first $r-1$ principal curvatures are equal, and the r-th is twice their common value.

Remark 2.

The assumption $r\ge 3$ is essential. For $r=2$, the normalized Casorati curvature ${\delta }_C(r-1)={\delta }_C\left(1\right)$ vanishes identically, making the inequalities trivial. We therefore assume $r\ge 3$ throughout, following Singh et al. [32], who established the general Casorati inequality for Riemannian maps under this assumption. However, for pointwise slant Riemannian maps, the identity ${\left(P^{\mathcal{R}}\right)}^2=-{cos}^2\theta ·I$ imposes an additional parity constraint. Indeed, for $\theta \in (0,\pi /2)$ we have $cos\theta \ne 0$, so the operator $J_0:=P^{\mathcal{R}}/cos\theta$ is well defined and satisfies

$$J_0^2={\displaystyle \frac{{\left(P^{\mathcal{R}}\right)}^2}{{cos}^2\theta }}=-I.$$

Hence, $J_0$ is an almost complex structure on the real vector space $Ran{F}_{\ast }$; since a real vector space admits an almost complex structure only if its dimension is even, it follows that $dim(Ran{F}_{\ast })=r$ must be even. Combined with the standing requirement $r\ge 3$, the smallest admissible value is $r=4$, so for proper pointwise slant maps into almost Hermitian manifolds the effective minimum rank is $r=4$. For Riemannian maps into almost contact metric manifolds with ${\xi }_2\in \Gamma (Ran{F}_{\ast })$, the same reasoning applies on $Ran{F}_{\ast }\cap ker{\eta }_2$, on which the slant condition is defined and which has dimension $r-1$; the parity constraint then forces $r-1$ to be even, so $r=3$ is the minimum admissible rank.

3. Pointwise Slant Riemannian Maps

With the necessary background established, we now introduce the notion of pointwise slant Riemannian maps and derive the key algebraic identities that will be used throughout the proofs of our main theorems.

3.1. Riemannian Map to an Almost Hermitian Manifold

Let $F:(M_1,g_1)\to (M_2,g_2,J_2)$ be a Riemannian map from a Riemannian manifold to an almost Hermitian manifold $M_2$, where the target $M_2$ carries an almost complex structure $J_2$. Using decomposition (2), for any $W\in \Gamma (Ran{F}_{\ast })$ we write

$$J_{2}W=P^{\mathcal{R}}W+Q^{\mathcal{R}}W,P^{\mathcal{R}}W\in \Gamma (Ran{F}_{\ast }),Q^{\mathcal{R}}W\in \Gamma {(Ran{F}_{\ast })}^{\perp }.$$

(13)

Here $P^{\mathcal{R}}:Ran{F}_{\ast }\to Ran{F}_{\ast }$ is the endomorphism that captures the component of $J_2$ tangential to $Ran{F}_{\ast }$, and $Q^{\mathcal{R}}W$, where $Q^{\mathcal{R}}:Ran{F}_{\ast }\to {(Ran{F}_{\ast })}^{\perp }$, is the complementary normal component.

Definition 4

([38]). A Riemannian map $F:(M_1,g_1)\to (M_2,g_2,J_2)$ to an almost Hermitian manifold is called pointwise slant if for each $p\in M_1$ there exists $\theta \left(p\right)\in [0,\pi /2]$ such that for every nonzero $X\in {(ker{F}_{\ast p})}^{\perp }$,

$$cos\theta \left(p\right)={\displaystyle \frac{\parallel P^{\mathcal{R}}\left(F_{\ast }X\right)\parallel }{\parallel F_{\ast }X\parallel }}.$$

The function $\theta :M_1\to [0,\pi /2]$ is the slant function.

When $\theta \equiv 0$ the map is invariant, i.e., $J_2(Ran{F}_{\ast })\subseteq Ran{F}_{\ast }$. When $\theta \equiv \pi /2$ the map is anti-invariant, i.e., $J_2(Ran{F}_{\ast })\subseteq {(Ran{F}_{\ast })}^{\perp }$. For general $\theta$ both components are nontrivial.

The following algebraic identities are established in [38] (Lemma 2.2), for all $W,W^{\prime }\in \Gamma (Ran{F}_{\ast })$,

$${\left(P^{\mathcal{R}}\right)}^2=-{cos}^2\theta ·I,g_2(P^{\mathcal{R}}W,P^{\mathcal{R}}{W}^{\prime })={cos}^2\theta ·g_2(W,W^{\prime }),g_2(Q^{\mathcal{R}}W,Q^{\mathcal{R}}{W}^{\prime })={sin}^2\theta ·g_2(W,W^{\prime }).$$

(14)

From these, Parseval’s identity applied to an orthonormal basis $\{F_{\ast }{e}_1,\dots ,F_{\ast }{e}_r\}$ of $Ran{F}_{\ast }$ gives the fundamental norm identity

$$\parallel P^{\mathcal{R}}{\parallel }^2:={\displaystyle \sum _{i,j=1}^r}{\left(g_2(F_{\ast }{e}_i,P^{\mathcal{R}}{F}_{\ast }{e}_j)\right)}^2=r{cos}^2\theta ,r=rankF.$$

(15)

3.2. Riemannian Map to an Almost Contact Metric Manifold

Let $F:(M_1,g_1)\to (M_2,g_2,{\varphi }_2,{\xi }_2,{\eta }_2)$ be a Riemannian map where the target $M_2$ carries an almost contact metric structure $({\varphi }_2,{\xi }_2,{\eta }_2)$.

Remark 3.

In any almost contact metric manifold $(M,\varphi ,\xi ,\eta ,g)$ the identity ${\varphi }^2=-I+\eta \otimes \xi$ holds [36], and in particular $\varphi \left(\xi \right)=0$ [35] (Chapter 4). Hence, ϕ annihilates ξ, so no slant angle can be defined in the ξ-direction. On the subspace $ker\eta \subset TM$, however, ${\varphi }^2=-I$ and $\parallel \varphi W\parallel =\parallel W\parallel$, so ϕ acts as an isometry and the angle between $\varphi \left(W\right)$ and any distribution is well-defined. The slant condition for Riemannian maps to almost contact metric manifolds is therefore restricted to $Ran{F}_{\ast }\cap ker{\eta }_2$. When ${\xi }_2\in \Gamma (Ran{F}_{\ast })$, one basis vector of $Ran{F}_{\ast }$ is annihilated by ${\varphi }_2$ and the slant is defined on the remaining $r-1$ directions; when ${\xi }_2\in \Gamma \left({(Ran{F}_{\ast })}^{\perp }\right)$, the condition $Ran{F}_{\ast }\cap ker{\eta }_2=Ran{F}_{\ast }$ holds and no basis vector is excluded.

Using decomposition (2), for any $W\in \Gamma (Ran{F}_{\ast }\cap ker{\eta }_2)$ we write

$${\varphi }_{2}W=P^{\mathcal{H}}W+Q^{\mathcal{H}}W,P^{\mathcal{H}}W\in \Gamma (Ran{F}_{\ast }),Q^{\mathcal{H}}W\in \Gamma \left({(Ran{F}_{\ast })}^{\perp }\right),$$

(16)

where $P^{\mathcal{H}}:Ran{F}_{\ast }\cap ker{\eta }_2\to Ran{F}_{\ast }$ captures the component of ${\varphi }_{2}W$ tangential to $Ran{F}_{\ast }$, and $Q^{\mathcal{H}}$ is the normal remainder. Explicitly,

$$P^{\mathcal{H}}:Ran{F}_{\ast }\cap ker{\eta }_2\to Ran{F}_{\ast },W\mapsto {\mathcal{V}}_{Ran}\left({\varphi }_{2}W\right),$$

where ${\mathcal{V}}_{Ran}$ denotes orthogonal projection onto $Ran{F}_{\ast }$ inside $T_{F\left(p\right)}{M}_2$.

Definition 5.

A Riemannian map $F:(M_1,g_1)\to (M_2,g_2,{\varphi }_2,{\xi }_2,{\eta }_2)$ to an almost contact metric manifold is called pointwise slant if for each $p\in M_1$ there exists $\theta \left(p\right)\in [0,\pi /2]$ such that for every nonzero $X\in {(ker{F}_{\ast p})}^{\perp }$ with ${\eta }_2\left(F_{\ast }X\right)=0$,

$$cos\theta \left(p\right)={\displaystyle \frac{\parallel P^{\mathcal{H}}\left(F_{\ast }X\right)\parallel }{\parallel F_{\ast }X\parallel }}.$$

The function $\theta :M_1\to [0,\pi /2]$ is the slant function.

When $\theta \equiv 0$ the map is invariant, ${\varphi }_2(Ran{F}_{\ast }\cap ker{\eta }_2)\subseteq Ran{F}_{\ast }$. When $\theta \equiv \pi /2$ the map is anti-invariant, ${\varphi }_2(Ran{F}_{\ast }\cap ker{\eta }_2)\subseteq {(Ran{F}_{\ast })}^{\perp }$. For general $\theta$ both components are nontrivial.

Since ${\varphi }_2^2=-I$ on $ker{\eta }_2$, the polarization argument of [38] (Lemma 2.2) gives, for all $W,W^{\prime }\in \Gamma (Ran{F}_{\ast }\cap ker{\eta }_2)$:

$$\begin{array}{cc}\hfill & {\left(P^{\mathcal{H}}\right)}^2=-{cos}^2\theta ·I,g_2(P^{\mathcal{H}}W,P^{\mathcal{H}}{W}^{\prime })={cos}^2\theta ·g_2(W,W^{\prime }),\hfill \\ \hfill & g_2(Q^{\mathcal{H}}W,Q^{\mathcal{H}}{W}^{\prime })={sin}^2\theta ·g_2(W,W^{\prime }).\hfill \end{array}$$

(17)

Write $r=rankF$. Since ${\varphi }_2$ annihilates ${\xi }_2$, the slant condition is active on the orthogonal complement of ${\xi }_2$ in $Ran{F}_{\ast }$, so the effective dimension drops by one when ${\xi }_2\in Ran{F}_{\ast }$. Parseval’s identity then gives the fundamental norm identity

$$\parallel P^{\mathcal{H}}{\parallel }^2:=\sum _{i,j}{\left(g_2(F_{\ast }{e}_i,P^{\mathcal{H}}{F}_{\ast }{e}_j)\right)}^2=\left\{\begin{array}{cc}(r-1){cos}^2\theta ,\hfill & {\xi }_2\in \Gamma (Ran{F}_{\ast }),\hfill \\ r{cos}^2\theta ,\hfill & {\xi }_2\in \Gamma \left({(Ran{F}_{\ast })}^{\perp }\right),\hfill \end{array}\right.$$

(18)

the sum running over an orthonormal basis of $Ran{F}_{\ast }\cap ker{\eta }_2$.

Proposition 2.

Let ${\mathcal{D}}_{\theta }$ denote the distribution on which the slant condition is imposed—$Ran{F}_{\ast }$ in the almost Hermitian case and $Ran{F}_{\ast }\cap ker{\eta }_2$ in the almost contact case—and set $d=dim{\mathcal{D}}_{\theta }$. Then the tangential projection operator P satisfies

$${\parallel P\parallel }^2=\sum _{i,j}{\left(g_2(F_{\ast }{e}_i,P{F}_{\ast }{e}_j)\right)}^2=d{cos}^2\theta$$

(19)

for any orthonormal basis $\left\{F_{\ast }{e}_i\right\}$ of ${\mathcal{D}}_{\theta }$. This is (15) in the almost Hermitian case ($d=r$) and (18) in the almost contact case, where $d=r-1$ or r according to whether ${\xi }_2$ lies in $Ran{F}_{\ast }$ or its orthogonal complement.

Proof. 

In both cases $g_2(P{F}_{\ast }{e}_j,P{F}_{\ast }{e}_j)={cos}^2\theta$ for each j, by (14) or (17). Summing over the d basis vectors by Parseval’s identity gives ${\parallel P\parallel }^2=d{cos}^2\theta$. □

Remark 4.

Proposition 2 encompasses the following classical special cases [23,34,48], where d denotes the dimension of the distribution ${\mathcal{D}}_{\theta }$ on which the slant condition is imposed.

(a) 

Invariant case ($\theta =0$): $cos\theta =1$, so $P=J$ (or $P=\varphi$ on $ker{\eta }_2$) and $Q=0$; the structure map preserves the chosen distribution entirely, ${\parallel P\parallel }^2=d$.

(b) 

Anti-invariant case ($\theta =\pi /2$): $cos\theta =0$, so $P=0$ and $Q=J$ (or $Q=\varphi$); the structure map sends every vector entirely out of the chosen distribution, ${\parallel P\parallel }^2=0$.

(c) 

Proper slant ($0<\theta <\pi /2$): both P and Q are nontrivial and ${\parallel P\parallel }^2=d{cos}^2\theta \in (0,d)$ interpolates continuously between the two extremes.

The Casorati inequalities in Section 4 therefore recover all previously known results for the invariant and anti-invariant cases simultaneously through the single parameter ${cos}^2\theta$.

With the fundamental properties of pointwise slant Riemannian maps now established, we proceed to derive our main Casorati inequalities.

4. Casorati Inequalities for Pointwise Slant Riemannian Maps

Let $F:(M_1^{m_1},g_1)\to (M_2^{m_2},g_2)$ be a Riemannian map of rank r, and let $\{Z_1,\dots ,Z_r\}$ be an orthonormal basis of ${(ker{F}_{\ast })}^{\perp }$ at $p\in M_1$. The scalar curvature of the horizontal space and of the range space are defined, respectively, by

$${\tau }^{\mathcal{H}}\left(p\right)=\sum _{1\le i<j\le r}K(Z_i\wedge Z_j)={\displaystyle \frac{1}{2}}{\displaystyle \sum _{i,j=1}^r}{g}_1(R^{M_1}(Z_i,Z_j)Z_j,Z_i),$$

(20)

$${\tau }^{\mathcal{R}}\left(p\right)={\displaystyle \frac{1}{2}}{\displaystyle \sum _{i,j=1}^r}{g}_2(R^{M_2}(F_{\ast }{Z}_i,F_{\ast }{Z}_j)F_{\ast }{Z}_j,F_{\ast }{Z}_i),$$

(21)

where $K(Z_i\wedge Z_j)$ denotes the sectional curvature of the plane spanned by $Z_i$ and $Z_j$. Their normalized forms are

$${\rho }^{\mathcal{H}}\left(p\right)={\displaystyle \frac{2{\tau }^{\mathcal{H}}\left(p\right)}{r(r-1)}},{\rho }^{\mathcal{R}}\left(p\right)={\displaystyle \frac{2{\tau }^{\mathcal{R}}\left(p\right)}{r(r-1)}}.$$

(22)

Let $\{V_{r+1},\dots ,V_{m_2}\}$ be an orthonormal basis of ${(Ran{F}_{\ast })}^{\perp }$. Define the components of the second fundamental form by

$$h_{ij}^{\alpha }=g_2((\nabla F_{\ast })(Z_i,Z_j),V_{\alpha }),i,j=1,\dots ,r,\alpha =r+1,\dots ,m_2.$$

(23)

The squared norm of the second fundamental form is

$${\parallel h\parallel }^2={\displaystyle \sum _{i,j=1}^r}{g}_2((\nabla F_{\ast })(Z_i,Z_j),(\nabla F_{\ast })(Z_i,Z_j))={\displaystyle \sum _{\alpha =r+1}^{m_2}}{\displaystyle \sum _{i,j=1}^r}{\left(h_{ij}^{\alpha }\right)}^2.$$

(24)

The Casorati curvature, introduced by Casorati [1] and systematically developed by Decu et al. [2], is defined as the normalized squared norm of the second fundamental form.

Definition 6

([1,2]). The Casorati curvature of the horizontal space is

$$C^{\mathcal{H}}={\displaystyle \frac{1}{r}}{\parallel h\parallel }^2={\displaystyle \frac{1}{r}}{\displaystyle \sum _{\alpha =r+1}^{m_2}}{\displaystyle \sum _{i,j=1}^r}{\left(h_{ij}^{\alpha }\right)}^2.$$

(25)

For a k-dimensional subspace $L\subset {(ker{F}_{\ast })}^{\perp }$ $(k\ge 2)$ with orthonormal basis $\{Z_1,\dots ,Z_k\}$, the Casorati curvature of L is

$$C\left(L\right)={\displaystyle \frac{1}{k}}{\displaystyle \sum _{\alpha =r+1}^{m_2}}{\displaystyle \sum _{i,j=1}^k}{\left(h_{ij}^{\alpha }\right)}^2.$$

(26)

Definition 7

([2]). The normalized Casorati curvatures are defined as

$${\delta }_C^{\mathcal{H}}(r-1)={\displaystyle \frac{1}{2}}{C}^{\mathcal{H}}+{\displaystyle \frac{r+1}{2r}}inf\{C\left(L\right):L\mathit{is}a\mathit{hyperplane}\mathit{of}{(ker{F}_{\ast })}^{\perp }\},$$

(27)

$${\widehat{\delta }}_C^{\mathcal{H}}(r-1)=2C^{\mathcal{H}}-{\displaystyle \frac{2r-1}{2r}}sup\{C\left(L\right):L\mathit{is}a\mathit{hyperplane}\mathit{of}{(ker{F}_{\ast })}^{\perp }\}.$$

(28)

The following notion, due to Singh et al. [32], generalizes classical quasi-umbilical submanifolds [2] to distributions in Riemannian maps.

Definition 8.

The horizontal distribution ${(ker{F}_{\ast })}^{\perp }$ of a Riemannian map $F:(M_1,g_1)\to (M_2,g_2)$ is called invariantly quasi-umbilical if there exist orthonormal bases $\{Z_1,\dots ,Z_r\}$ of ${(ker{F}_{\ast })}^{\perp }$ and $\{V_{r+1},\dots ,V_{m_2}\}$ of ${(Ran{F}_{\ast })}^{\perp }$ such that the components $h_{ij}^{\alpha }=g_2((\nabla F_{\ast })(Z_i,Z_j),V_{\alpha })$ satisfy

(i) 

$h_{ij}^{\alpha }=0$ for all $i\ne j$ and all $\alpha =r+1,\dots ,m_2$;

(ii) 

$h_{11}^{\alpha }=h_{22}^{\alpha }=\cdots =h_{r-1,r-1}^{\alpha }$ for all $\alpha =r+1,\dots ,m_2$.

Remark 5.

Condition (i) ensures the second fundamental form is diagonal in suitable bases, while condition (ii) requires the first $r-1$ principal curvatures to be equal. The equality case in Theorem 2 below corresponds to the sharper requirement

$$h_{11}^{\alpha }=h_{22}^{\alpha }=\cdots =h_{r-1,r-1}^{\alpha }={\displaystyle \frac{1}{2}}{h}_{rr}^{\alpha },h_{ij}^{\alpha }=0\mathrm{for}i\ne j,$$

for all $\alpha =r+1,\dots ,m_2$.

4.1. Casorati Inequalities for Pointwise Slant Riemannian Maps to Generalized Complex Space Forms

Our goal in this section is to derive explicit Casorati inequalities for pointwise slant Riemannian maps, specializing to the cases where the base manifold is a generalized complex space form (Theorem 2). The key ingredients are the Gauss Equation (8) and the algebraic optimization lemma, Lemma 1.

Recall from Definition 2 that a generalized complex space form $M_2(c_1,c_2)$ is characterized by the curvature condition (9). We now present our first main result for pointwise slant Riemannian maps.

Theorem 2.

Let $F:(M_1^{m_1},g_1)\to (M_2^{m_2}(c_1,c_2),g_2,J_2)$ be a pointwise slant Riemannian map from a Riemannian manifold to a generalized complex space form with rank $r\ge 3$ and slant function θ. Then

$${\rho }^{\mathcal{H}}\le {\delta }_C^{\mathcal{H}}(r-1)+c_1+{\displaystyle \frac{3c_2}{r(r-1)}}{\parallel P^{\mathcal{R}}\parallel }^2={\delta }_C^{\mathcal{H}}(r-1)+c_1+{\displaystyle \frac{3c_2}{r-1}}{cos}^2\theta ,$$

(29)

and

$${\rho }^{\mathcal{H}}\le {\widehat{\delta }}_C^{\mathcal{H}}(r-1)+c_1+{\displaystyle \frac{3c_2}{r(r-1)}}{\parallel P^{\mathcal{R}}\parallel }^2={\widehat{\delta }}_C^{\mathcal{H}}(r-1)+c_1+{\displaystyle \frac{3c_2}{r-1}}{cos}^2\theta .$$

(30)

Equality holds in (29) or (30) at a point $p\in M_1$ if and only if, for suitable orthonormal bases, the components of the second fundamental form satisfy

$$h_{11}^{\alpha }=h_{22}^{\alpha }=\cdots =h_{r-1,r-1}^{\alpha }={\displaystyle \frac{1}{2}}{h}_{rr}^{\alpha },h_{ij}^{\alpha }=0\mathrm{for}i\ne j,$$

for all $\alpha =r+1,\dots ,m_2$.

Proof. 

We compute ${\rho }^{\mathcal{R}}$ using the curvature Formula (9) for generalized complex space forms. Using Equation (21) and the orthonormal basis $\{F_{\ast }{Z}_1,\dots ,F_{\ast }{Z}_r\}$ of $Ran{F}_{\ast }$, we have

$$\begin{array}{ccc}\hfill 2{\tau }^{\mathcal{R}}& =& {\displaystyle \sum _{i,j=1}^r}{g}_2(R^{M_2}(F_{\ast }{Z}_i,F_{\ast }{Z}_j)F_{\ast }{Z}_j,F_{\ast }{Z}_i)\hfill \\ & =& {\displaystyle \sum _{i,j=1}^r}{c}_1\{g_2(F_{\ast }{Z}_j,F_{\ast }{Z}_j)g_2(F_{\ast }{Z}_i,F_{\ast }{Z}_i)-g_2(F_{\ast }{Z}_i,F_{\ast }{Z}_j)g_2(F_{\ast }{Z}_j,F_{\ast }{Z}_i)\}\hfill \\ & & +{\displaystyle \sum _{i,j=1}^r}{c}_2\{g_2(F_{\ast }{Z}_i,J_2{F}_{\ast }{Z}_j)g_2(J_2{F}_{\ast }{Z}_j,F_{\ast }{Z}_i)\hfill \\ & & -g_2(F_{\ast }{Z}_j,J_2{F}_{\ast }{Z}_j)g_2(J_2{F}_{\ast }{Z}_i,F_{\ast }{Z}_i)\hfill \\ & & +2g_2(F_{\ast }{Z}_i,J_2{F}_{\ast }{Z}_j)g_2(J_2{F}_{\ast }{Z}_j,F_{\ast }{Z}_i)\}.\hfill \end{array}$$

Using the orthonormality $g_2(F_{\ast }{Z}_i,F_{\ast }{Z}_j)={\delta }_{ij}$, the first sum simplifies as

$${\displaystyle \sum _{i,j=1}^r}{c}_1\{1·1-{\delta }_{ij}{\delta }_{ij}\}=c_1{\displaystyle \sum _{i,j=1}^r}(1-{\delta }_{ij})=c_1(r^2-r)=c_{1}r(r-1).$$

For the second sum, we use the decomposition (13). Since $J_2{F}_{\ast }{Z}_j=P^{\mathcal{R}}{F}_{\ast }{Z}_j+Q^{\mathcal{R}}{F}_{\ast }{Z}_j$ with $P^{\mathcal{R}}{F}_{\ast }{Z}_j\in Ran{F}_{\ast }$ and $Q^{\mathcal{R}}{F}_{\ast }{Z}_j\in {(Ran{F}_{\ast })}^{\perp }$, and using the orthogonality of these spaces, we have

$$g_2(F_{\ast }{Z}_i,J_2{F}_{\ast }{Z}_j)=g_2(F_{\ast }{Z}_i,P^{\mathcal{R}}{F}_{\ast }{Z}_j).$$

Moreover, $g_2(F_{\ast }{Z}_j,J_2{F}_{\ast }{Z}_j)=g_2(F_{\ast }{Z}_j,P^{\mathcal{R}}{F}_{\ast }{Z}_j)$, but since $J_2$ is an almost complex structure, $g_2(J_{2}X,X)=0$ for all X, which implies $g_2(F_{\ast }{Z}_j,J_2{F}_{\ast }{Z}_j)=0$. □

Therefore, the second sum becomes

$$\begin{array}{cc}\hfill & {\displaystyle \sum _{i,j=1}^r}{c}_2\{g_2(F_{\ast }{Z}_i,P^{\mathcal{R}}{F}_{\ast }{Z}_j)g_2(P^{\mathcal{R}}{F}_{\ast }{Z}_j,F_{\ast }{Z}_i)-0+2g_2(F_{\ast }{Z}_i,P^{\mathcal{R}}{F}_{\ast }{Z}_j)g_2(P^{\mathcal{R}}{F}_{\ast }{Z}_j,F_{\ast }{Z}_i)\}\hfill \\ \hfill & =3c_2{\displaystyle \sum _{i,j=1}^r}{\left(g_2(F_{\ast }{Z}_i,P^{\mathcal{R}}{F}_{\ast }{Z}_j)\right)}^2=3c_2{\parallel P^{\mathcal{R}}\parallel }^2,\hfill \end{array}$$

where we used definition (15).

Combining these results, we obtain

$$2{\tau }^{\mathcal{R}}=r(r-1)c_1+3c_2{\parallel P^{\mathcal{R}}\parallel }^2.$$

(31)

Dividing by $r(r-1)$ yields

$${\rho }^{\mathcal{R}}=c_1+{\displaystyle \frac{3c_2}{r(r-1)}}{\parallel P^{\mathcal{R}}\parallel }^2.$$

(32)

On the other hand, setting $X=W=Z_i$ and $Y=Z=Z_j$ in the Gauss Equation (8) and summing over $i,j=1,\dots ,r$, we obtain

$$\begin{array}{ccc}\hfill 2{\tau }^{\mathcal{R}}& =& 2{\tau }^{\mathcal{H}}+{\displaystyle \sum _{i,j=1}^r}{g}_2\left((\nabla F_{\ast })(Z_i,Z_i),(\nabla F_{\ast })(Z_j,Z_j)\right)-{\displaystyle \sum _{i,j=1}^r}{g}_2\left((\nabla F_{\ast })(Z_i,Z_j),(\nabla F_{\ast })(Z_i,Z_j)\right)\hfill \\ & =& 2{\tau }^{\mathcal{H}}+{\parallel h\parallel }^2-{\parallel trh\parallel }^2,\hfill \end{array}$$

where $trh={\sum }_{i=1}^r(\nabla F_{\ast })(Z_i,Z_i)$, the second sum equals $g_2\left({\sum }_i(\nabla F_{\ast })(Z_i,Z_i),{\sum }_j(\nabla F_{\ast })(Z_j,Z_j)\right)={\parallel trh\parallel }^2$, and the first sum equals ${\parallel h\parallel }^2$ by definition (24). Rearranging yields

$$2{\tau }^{\mathcal{H}}=2{\tau }^{\mathcal{R}}-{\parallel h\parallel }^2+{\parallel trh\parallel }^2.$$

(33)

Consider the quadratic polynomial

$$\mathcal{P}={\displaystyle \frac{1}{2}}r(r-1)C^{\mathcal{H}}+{\displaystyle \frac{1}{2}}(r^2-1)C\left(L\right)+2{\tau }^{\mathcal{R}}-2{\tau }^{\mathcal{H}},$$

where L is the hyperplane of ${(ker{F}_{\ast })}^{\perp }$ spanned by $\{Z_1,\dots ,Z_{r-1}\}$. We expand $\mathcal{P}$ in terms of the components $h_{ij}^{\alpha }$. By the definitions (25) and (26), for each fixed $\alpha$,

$${\displaystyle \frac{1}{2}}r(r-1)C^{\mathcal{H}}{|}_{\alpha }={\displaystyle \frac{r-1}{2}}{\displaystyle \sum _{i,j=1}^r}{\left(h_{ij}^{\alpha }\right)}^2,{\displaystyle \frac{1}{2}}(r^2-1)C\left(L\right){|}_{\alpha }={\displaystyle \frac{r+1}{2}}{\displaystyle \sum _{i,j=1}^{r-1}}{\left(h_{ij}^{\alpha }\right)}^2,$$

while by (33),

$$2{\tau }^{\mathcal{R}}-2{\tau }^{\mathcal{H}}={\parallel h\parallel }^2-{\parallel trh\parallel }^2={\displaystyle \sum _{\alpha =r+1}^{m_2}}\left[{\displaystyle \sum _{i,j=1}^r}{\left(h_{ij}^{\alpha }\right)}^2-{\left({\displaystyle \sum _{i=1}^r}{h}_{ii}^{\alpha }\right)}^2\right].$$

Adding these three contributions and using ${\displaystyle \frac{r-1}{2}}+1={\displaystyle \frac{r+1}{2}}$ gives

$$\mathcal{P}={\displaystyle \sum _{\alpha =r+1}^{m_2}}\left[{\displaystyle \frac{r+1}{2}}{\displaystyle \sum _{i,j=1}^r}{\left(h_{ij}^{\alpha }\right)}^2+{\displaystyle \frac{r+1}{2}}{\displaystyle \sum _{i,j=1}^{r-1}}{\left(h_{ij}^{\alpha }\right)}^2-{\left({\displaystyle \sum _{i=1}^r}{h}_{ii}^{\alpha }\right)}^2\right].$$

We now separate, in each squared sum, the diagonal entries $(i=j)$, the last row and column $(j=r,i<r$, together with $i=r,j<r)$, and the remaining off-diagonal entries $(i,j\le r-1)$, using the symmetry $h_{ij}^{\alpha }=h_{ji}^{\alpha }$ and the identity ${\left({\sum }_i{h}_{ii}^{\alpha }\right)}^2={\sum }_i{\left(h_{ii}^{\alpha }\right)}^2+2{\sum }_{1\le i<j\le r}{h}_{ii}^{\alpha }{h}_{jj}^{\alpha }$. Collecting like terms, the resulting coefficients are r for ${\left(h_{ii}^{\alpha }\right)}^2$ with $i\le r-1$, namely ${\displaystyle \frac{r+1}{2}}+{\displaystyle \frac{r+1}{2}}-1=r$; the value ${\displaystyle \frac{r-1}{2}}$ for ${\left(h_{rr}^{\alpha }\right)}^2$, namely ${\displaystyle \frac{r+1}{2}}-1$; the value $-2$ for the products $h_{ii}^{\alpha }{h}_{jj}^{\alpha }$; the value $(r+1)$ for ${\left(h_{ir}^{\alpha }\right)}^2$ with $i\le r-1$, arising as $2·{\displaystyle \frac{r+1}{2}}$ from the first sum; and $2(r+1)$ for ${\left(h_{ij}^{\alpha }\right)}^2$ with $i<j\le r-1$, arising as $2·{\displaystyle \frac{r+1}{2}}+2·{\displaystyle \frac{r+1}{2}}$. This yields

$$\begin{array}{cc}\hfill \mathcal{P}& ={\displaystyle \sum _{\alpha =r+1}^{m_2}}[r{\displaystyle \sum _{i=1}^{r-1}}{\left(h_{ii}^{\alpha }\right)}^2+{\displaystyle \frac{r-1}{2}}{\left(h_{rr}^{\alpha }\right)}^2-2\sum _{1\le i<j\le r}{h}_{ii}^{\alpha }{h}_{jj}^{\alpha }\hfill \\ \hfill & +(r+1){\displaystyle \sum _{i=1}^{r-1}}{\left(h_{ir}^{\alpha }\right)}^2+2(r+1)\sum _{1\le i<j\le r-1}{\left(h_{ij}^{\alpha }\right)}^2].\hfill \end{array}$$

(34)

The off-diagonal coefficients in (34) are positive: $(r+1)>0$ for the $h_{ir}^{\alpha }$-terms, and $(r-1)+(r+1)-2=2(r-1)>0$ for the $h_{ij}^{\alpha }$-terms ($i<j\le r-1$), since $r\ge 3$. For each fixed $\alpha$, the diagonal part is exactly the quadratic form in Lemma 1 with ${\lambda }_1=r$ and ${\lambda }_2=(r-1)/2$, subject to the constraint $h_{11}^{\alpha }+\cdots +h_{rr}^{\alpha }=k^{\alpha }$. These values satisfy the hypothesis of Lemma 1: since $r\ge 3$ we have ${\lambda }_1-r+2=2>0$, and

$${\displaystyle \frac{r-1}{{\lambda }_1-r+2}}={\displaystyle \frac{r-1}{2}}={\lambda }_2,$$

so that ${\lambda }_1,{\lambda }_2>0$ with ${\lambda }_2=(r-1)/({\lambda }_1-r+2)$ as required (cf. Remark 1). By Lemma 1, the constrained minimum is 0, so $\mathcal{P}\ge 0$. Rearranging and dividing by $r(r-1)$, then taking the infimum over all hyperplanes $L\subset {(ker{F}_{\ast })}^{\perp }$, we get

$${\rho }^{\mathcal{H}}\le {\delta }_C^{\mathcal{H}}(r-1)+{\rho }^{\mathcal{R}}.$$

Similarly, considering $\widehat{\mathcal{P}}=2r(r-1)C^{\mathcal{H}}-{\displaystyle \frac{1}{2}}(r-1)(2r-1)C\left(L\right)+2{\tau }^{\mathcal{R}}-2{\tau }^{\mathcal{H}}$ and taking the supremum gives ${\rho }^{\mathcal{H}}\le {\widehat{\delta }}_C^{\mathcal{H}}(r-1)+{\rho }^{\mathcal{R}}$. Substituting (32) and using ${\parallel P^{\mathcal{R}}\parallel }^2=r{cos}^2\theta$ from (15), we obtain the desired inequalities (29) and (30).

Equality holds in (29) if and only if $\mathcal{P}=0$. From (34), this requires $h_{ij}^{\alpha }=0$ for all $i\ne j$ (since the off-diagonal terms have strictly positive coefficients), and the constrained minimum of the diagonal quadratic form equals zero, which by Lemma 1 holds if and only if there exist orthonormal bases of ${(ker{F}_{\ast })}^{\perp }$ and ${(Ran{F}_{\ast })}^{\perp }$, with $Z_r$ aligned to the distinguished principal direction so that $L=span\{Z_1,\dots ,Z_{r-1}\}$ attains the infimum, in which the components of the second fundamental form satisfy

$$h_{11}^{\alpha }=h_{22}^{\alpha }=\cdots =h_{r-1,r-1}^{\alpha }={\displaystyle \frac{1}{2}}{h}_{rr}^{\alpha },h_{ij}^{\alpha }=0(i\ne j),$$

for all $\alpha =r+1,\dots ,m_2$.

The significance of Theorem 2 is that it makes the dependence on the slant angle $\theta$ completely explicit through the factor ${cos}^2\theta$. As $\theta$ varies from 0 to $\pi /2$, the term ${\displaystyle \frac{3c_2}{r-1}}{cos}^2\theta$ varies from ${\displaystyle \frac{3c_2}{r-1}}$ to 0, showing precisely how the Casorati bound interpolates between these extremes. We now specialize to particular types of generalized complex space forms.

Corollary 1.

Let $F:(M_1^{m_1},g_1)\to (M_2^{m_2},g_2)$ be a pointwise slant Riemannian map with rank $r\ge 3$ and slant function θ.

1. 

If $M_2\left(c\right)$ is a real space form ($c_1=c,c_2=0$), then

$${\rho }^{\mathcal{H}}\le {\delta }_C^{\mathcal{H}}(r-1)+c,{\rho }^{\mathcal{H}}\le {\widehat{\delta }}_C^{\mathcal{H}}(r-1)+c.$$

(35)

2. 

If $M_2\left(c\right)$ is a complex space form ($c_1=c_2=c/4$), then

$${\rho }^{\mathcal{H}}\le {\delta }_C^{\mathcal{H}}(r-1)+{\displaystyle \frac{c}{4}}\left(1+{\displaystyle \frac{3}{r-1}}{cos}^2\theta \right),$$

(36)

$${\rho }^{\mathcal{H}}\le {\widehat{\delta }}_C^{\mathcal{H}}(r-1)+{\displaystyle \frac{c}{4}}\left(1+{\displaystyle \frac{3}{r-1}}{cos}^2\theta \right).$$

(37)

3. 

If $M_2\left(c\right)$ is a real Kähler space form ($c_1={\displaystyle \frac{c+3\alpha }{4}},c_2={\displaystyle \frac{c-\alpha }{4}}$), then

$${\rho }^{\mathcal{H}}\le {\delta }_C^{\mathcal{H}}(r-1)+{\displaystyle \frac{c+3\alpha }{4}}+{\displaystyle \frac{3(c-\alpha )}{4(r-1)}}{cos}^2\theta ,$$

(38)

$${\rho }^{\mathcal{H}}\le {\widehat{\delta }}_C^{\mathcal{H}}(r-1)+{\displaystyle \frac{c+3\alpha }{4}}+{\displaystyle \frac{3(c-\alpha )}{4(r-1)}}{cos}^2\theta .$$

(39)

Proof. 

These follow immediately by substituting the appropriate values of $c_1$ and $c_2$ into Theorem 2. For real space forms, $c_2=0$ eliminates the theta-dependent term entirely, showing that the inequality is independent of $\theta$ in this case. For complex and Kähler space forms, the slant dependence enters through the $c_2$ coefficient. □

We now give a complete characterization of totally geodesic pointwise slant Riemannian maps into generalized complex space forms in terms of the equality cases of Theorem 2.

Corollary 2.

Let $F:(M_1^{m_1},g_1)\to (M_2^{m_2}(c_1,c_2),g_2,J_2)$ be a pointwise slant Riemannian map with rank $r\ge 3$ and slant function θ. Then F is totally geodesic if and only if equality holds simultaneously in both (29) and (30).

Proof. 

$(\Rightarrow )$ If $\nabla F_{\ast }\equiv 0$, then $h_{ij}^{\alpha }=0$ for all $i,j,\alpha$ by (23), so ${\parallel h\parallel }^2=0$ by (24). Hence, $C^{\mathcal{H}}=0$ by (25) and ${\delta }_C^{\mathcal{H}}(r-1)={\widehat{\delta }}_C^{\mathcal{H}}(r-1)=0$ by (27) and (28). Both inequalities of Theorem 2 then reduce to equalities ${\rho }^{\mathcal{H}}=c_1+{\displaystyle \frac{3c_2}{r-1}}{cos}^2\theta$.

$(\Leftarrow )$ Suppose both equalities hold simultaneously. Then ${\delta }_C^{\mathcal{H}}(r-1)={\widehat{\delta }}_C^{\mathcal{H}}(r-1)$, which by (27) and (28) is equivalent to

$${\displaystyle \frac{r+1}{2r}}infC\left(L\right)+{\displaystyle \frac{2r-1}{2r}}supC\left(L\right)={\displaystyle \frac{3}{2}}{C}^{\mathcal{H}}.$$

(40)

By the equality analysis of Theorem 2 (cf. Remark 5), there exist orthonormal bases in which $h_{ij}^{\alpha }=0$$(i\ne j)$, $h_{11}^{\alpha }=\cdots =h_{r-1,r-1}^{\alpha }=a^{\alpha }$ and $h_{rr}^{\alpha }=2a^{\alpha }$ for all $\alpha$. Setting $S:={\sum }_{\alpha }{\left(a^{\alpha }\right)}^2$, the diagonal form gives $C^{\mathcal{H}}={\displaystyle \frac{1}{r}}{\sum }_{\alpha }[(r-1)+4]{\left(a^{\alpha }\right)}^2={\displaystyle \frac{r+3}{r}}S$. The hyperplane curvature $C\left(L\right)$, however, is not constant, the hyperplane omitting the distinguished direction $Z_r$ (where $h_{rr}^{\alpha }=2a^{\alpha }$) gives $C\left(L\right)=S$, while for a coordinate hyperplane retaining $Z_r$ gives $C\left(L\right)={\displaystyle \frac{1}{r-1}}{\sum }_{\alpha }[(r-2)+4]{\left(a^{\alpha }\right)}^2={\displaystyle \frac{r+2}{r-1}}S>S$. Hence, $infC\left(L\right)=S$ and $supC\left(L\right)={\displaystyle \frac{r+2}{r-1}}S$. Substituting these together with $C^{\mathcal{H}}={\displaystyle \frac{r+3}{r}}S$ into (40) and clearing the denominator $2r(r-1)$ reduces the identity to

$${\displaystyle \frac{3(r-2)}{2r(r-1)}}S=0.$$

Since $r\ge 3$, the coefficient is strictly positive, so $S={\sum }_{\alpha }{\left(a^{\alpha }\right)}^2=0$; thus every $a^{\alpha }=0$, all $h_{ij}^{\alpha }=0$, and $\nabla F_{\ast }\equiv 0$. Therefore, F is totally geodesic. □

Remark 6.

When $c_2>0$ (which holds for complex space forms with $c>0$, Sasakian space forms with $c>1$, and Kenmotsu space forms with $c>-1$), the upper bound in Theorem 2 is monotonically decreasing in θ: as the slant angle increases from 0 to $\pi /2$, the term ${\displaystyle \frac{3c_2}{r-1}}{cos}^2\theta$ decreases from ${\displaystyle \frac{3c_2}{r-1}}$ to 0. Geometrically, anti-invariant maps ($\theta =\pi /2$) satisfy tighter Casorati bounds than invariant maps ($\theta =0$) when $c_2>0$: the more the almost complex structure rotates vectors out of the range, the more constrained the extrinsic curvature becomes. Conversely, when $c_2<0$, the bound increases with θ and invariant maps have the tightest bounds. For real space forms ($c_2=0$), and likewise at the boundary value $c=1$ of the Sasakian family (where $c_2=(c-1)/4=0$, the case of the standard Sasakian sphere), the bound is independent of θ, reflecting the absence of a genuine slant contribution. The same monotonicity applies to the Sasakian case (Theorem 3 below).

Remark 7.

Corollary 1 demonstrates important consistency with known results.

(a) 

When $\theta =0$ (invariant case, $cos\theta =1$), inequality (36) becomes

$${\rho }^{\mathcal{H}}\le {\delta }_C^{\mathcal{H}}(r-1)+{\displaystyle \frac{c}{4}}\left(1+{\displaystyle \frac{3}{r-1}}\right)={\delta }_C^{\mathcal{H}}(r-1)+{\displaystyle \frac{c}{4}}·{\displaystyle \frac{r+2}{r-1}},$$

which agrees with the invariant Riemannian map inequalities in Lee et al. [28].

(b) 

When $\theta =\pi /2$ (anti-invariant case, $cos\theta =0$), inequality (36) reduces to

$${\rho }^{\mathcal{H}}\le {\delta }_C^{\mathcal{H}}(r-1)+{\displaystyle \frac{c}{4}},$$

which is precisely the anti-invariant inequality, again consistent with [28].

(c) 

For real space forms, the inequality (35) is independent of θ, which is geometrically natural since real space forms have no almost complex structure to measure slant angle against. This shows that the slant condition is vacuous in the case of real space forms.

4.2. Casorati Inequalities for Pointwise Slant Riemannian Maps to Generalized Sasakian Space Forms

We now turn to pointwise slant Riemannian maps into generalized Sasakian space forms. The analysis is similar but involves an additional parameter $c_3$ related to the Reeb vector field, and we must consider two cases depending on whether the Reeb vector field ${\xi }_2$ lies in the range space or its orthogonal complement.

Theorem 3.

Let $F:(M_1^{m_1},g_1)\to (M_2^{m_2}(c_1,c_2,c_3),{\varphi }_2,{\xi }_2,{\eta }_2,g_2)$ be a pointwise slant Riemannian map from a Riemannian manifold to a generalized Sasakian space form with rank $r\ge 3$ and slant function θ. Then

$${\rho }^{\mathcal{H}}\le \left\{\begin{array}{cc}{\delta }_C^{\mathcal{H}}(r-1)+c_1+{\displaystyle \frac{3c_2}{r(r-1)}}{\parallel P^{\mathcal{H}}\parallel }^2-{\displaystyle \frac{2}{r}}{c}_3,\hfill & \mathit{if}{\xi }_2\in \Gamma (Ran{F}_{\ast }),\hfill \\ {\delta }_C^{\mathcal{H}}(r-1)+c_1+{\displaystyle \frac{3c_2}{r(r-1)}}{\parallel P^{\mathcal{H}}\parallel }^2,\hfill & \mathit{if}{\xi }_2\in \Gamma \left({(Ran{F}_{\ast })}^{\perp }\right),\hfill \end{array}\right.$$

(41)

and

$${\rho }^{\mathcal{H}}\le \left\{\begin{array}{cc}{\widehat{\delta }}_C^{\mathcal{H}}(r-1)+c_1+{\displaystyle \frac{3c_2}{r(r-1)}}{\parallel P^{\mathcal{H}}\parallel }^2-{\displaystyle \frac{2}{r}}{c}_3,\hfill & \mathit{if}{\xi }_2\in \Gamma (Ran{F}_{\ast }),\hfill \\ {\widehat{\delta }}_C^{\mathcal{H}}(r-1)+c_1+{\displaystyle \frac{3c_2}{r(r-1)}}{\parallel P^{\mathcal{H}}\parallel }^2,\hfill & \mathit{if}{\xi }_2\in \Gamma \left({(Ran{F}_{\ast })}^{\perp }\right).\hfill \end{array}\right.$$

(42)

By (18), $\parallel P^{\mathcal{H}}{\parallel }^2=(r-1){cos}^2\theta$ when ${\xi }_2\in \Gamma (Ran{F}_{\ast })$ and $\parallel P^{\mathcal{H}}{\parallel }^2=r{cos}^2\theta$ when ${\xi }_2\in \Gamma \left({(Ran{F}_{\ast })}^{\perp }\right)$, so these become

$${\rho }^{\mathcal{H}}\le \left\{\begin{array}{cc}{\delta }_C^{\mathcal{H}}(r-1)+c_1+{\displaystyle \frac{3c_2}{r}}{cos}^2\theta -{\displaystyle \frac{2}{r}}{c}_3,\hfill & \mathit{if}{\xi }_2\in \Gamma (Ran{F}_{\ast }),\hfill \\ {\delta }_C^{\mathcal{H}}(r-1)+c_1+{\displaystyle \frac{3c_2}{r-1}}{cos}^2\theta ,\hfill & \mathit{if}{\xi }_2\in \Gamma \left({(Ran{F}_{\ast })}^{\perp }\right),\hfill \end{array}\right.$$

(43)

and similarly for the ${\widehat{\delta }}_C^{\mathcal{H}}$. The equality conditions are the same as in Theorem 2.

Proof. 

The proof parallels that of Theorem 2 but uses the curvature Formula (10) for generalized Sasakian space forms. Computing $2{\tau }^{\mathcal{R}}$ using an orthonormal basis $\{F_{\ast }{Z}_1,\dots ,F_{\ast }{Z}_r\}$ of $Ran{F}_{\ast }$, we obtain contributions from the three terms in (10)

$$\begin{array}{ccc}\hfill 2{\tau }^{\mathcal{R}}& =& r(r-1)c_1+3c_2\parallel P^{\mathcal{H}}{\parallel }^2+c_3{\displaystyle \sum _{i,j=1}^r}\{{\eta }_2\left(F_{\ast }{Z}_i\right){\eta }_2\left(F_{\ast }{Z}_j\right)g_2(F_{\ast }{Z}_j,F_{\ast }{Z}_i)\hfill \\ & & -{\eta }_2\left(F_{\ast }{Z}_j\right){\eta }_2\left(F_{\ast }{Z}_j\right)g_2(F_{\ast }{Z}_i,F_{\ast }{Z}_i)+g_2(F_{\ast }{Z}_i,F_{\ast }{Z}_j){\eta }_2\left(F_{\ast }{Z}_j\right)g_2({\xi }_2,F_{\ast }{Z}_i)\hfill \\ & & -g_2(F_{\ast }{Z}_j,F_{\ast }{Z}_j){\eta }_2\left(F_{\ast }{Z}_i\right)g_2({\xi }_2,F_{\ast }{Z}_i)\}.\hfill \end{array}$$

The first two terms match the generalized complex case. For the third term, we consider two cases:

Case 1 If ${\xi }_2\in \Gamma (Ran{F}_{\ast })$, we include ${\xi }_2$ in the orthonormal basis by setting $F_{\ast }{Z}_r={\xi }_2$, so that

$${\eta }_2\left(F_{\ast }{Z}_i\right)=g_2({\xi }_2,F_{\ast }{Z}_i)={\delta }_{ir},g_2(F_{\ast }{Z}_i,F_{\ast }{Z}_j)={\delta }_{ij}.$$

Writing the $c_3$-bracket of (10) as $T_1+T_2+T_3+T_4$, we note that the four terms evaluate individually as follows. For the first term, ${\delta }_{ij}$ forces $i=j$, whence

$$T_1=\sum _{i,j}{\eta }_2\left(F_{\ast }{Z}_i\right){\eta }_2\left(F_{\ast }{Z}_j\right)g_2(F_{\ast }{Z}_j,F_{\ast }{Z}_i)=\sum _{i,j}{\delta }_{ir}{\delta }_{jr}{\delta }_{ij}=\sum _i{\delta }_{ir}^2=1.$$

For the second term, $g_2(F_{\ast }{Z}_i,F_{\ast }{Z}_i)=1$ and ${\sum }_j{\delta }_{jr}^2=1$, so

$$T_2=-\sum _{i,j}{\eta }_2{\left(F_{\ast }{Z}_j\right)}^2{g}_2(F_{\ast }{Z}_i,F_{\ast }{Z}_i)=-\sum _{i,j}{\delta }_{jr}^2=-\sum _{i}1=-r.$$

For the third term, $g_2(F_{\ast }{Z}_i,F_{\ast }{Z}_j)={\delta }_{ij}$ again forces $i=j$, giving

$$T_3=\sum _{i,j}{g}_2(F_{\ast }{Z}_i,F_{\ast }{Z}_j){\eta }_2\left(F_{\ast }{Z}_j\right)g_2({\xi }_2,F_{\ast }{Z}_i)=\sum _{i,j}{\delta }_{ij}{\delta }_{jr}{\delta }_{ir}=\sum _i{\delta }_{ir}^2=1.$$

For the fourth term, $g_2(F_{\ast }{Z}_j,F_{\ast }{Z}_j)=1$ and ${\sum }_i{\delta }_{ir}^2=1$, so

$$T_4=-\sum _{i,j}{g}_2(F_{\ast }{Z}_j,F_{\ast }{Z}_j){\eta }_2\left(F_{\ast }{Z}_i\right)g_2({\xi }_2,F_{\ast }{Z}_i)=-\sum _{i,j}{\delta }_{ir}^2=-\sum _{j}1=-r.$$

Hence, the total $c_3$-contribution to $2{\tau }^{\mathcal{R}}$ is

$$c_3(T_1+T_2+T_3+T_4)=c_3(1-r+1-r)=-2c_3(r-1),$$

giving an additional term $-2c_3(r-1)/\left(r(r-1)\right)=-2c_3/r$ in ${\rho }^{\mathcal{R}}$.

Case 2 If ${\xi }_2\in \Gamma \left({(Ran{F}_{\ast })}^{\perp }\right)$, then $g_2({\xi }_2,F_{\ast }{Z}_i)=0$ for all i, so the third term vanishes and we obtain the same formula as in the generalized complex case.

Substituting the above expression for ${\rho }^{\mathcal{R}}$, namely $c_1+{\displaystyle \frac{3c_2}{r(r-1)}}{\parallel P^{\mathcal{H}}\parallel }^2-{\displaystyle \frac{2}{r}}{c}_3$ when ${\xi }_2\in \Gamma (Ran{F}_{\ast })$, and $c_1+{\displaystyle \frac{3c_2}{r(r-1)}}{\parallel P^{\mathcal{H}}\parallel }^2$ when ${\xi }_2\in \Gamma \left({(Ran{F}_{\ast })}^{\perp }\right)$, into the inequality ${\rho }^{\mathcal{H}}\le {\delta }_C^{\mathcal{H}}(r-1)+{\rho }^{\mathcal{R}}$ obtained in the proof of Theorem 2, yields (41) and (42). The proof for the ${\widehat{\delta }}_C^{\mathcal{H}}$ is identical. □

We now specialize Theorem 3 to the classical space form cases.

Corollary 3.

Let $F:(M_1^{m_1},g_1)\to (M_2^{m_2}(c_1,c_2,c_3),{\varphi }_2,{\xi }_2,{\eta }_2,g_2)$ be a pointwise slant Riemannian map to a generalized Sasakian space form with rank $r\ge 3$ and slant function θ.

(1) 

If $M_2\left(c\right)$ is a Sasakian space form ($c_1={\displaystyle \frac{c+3}{4}},c_2=c_3={\displaystyle \frac{c-1}{4}}$), then

$${\rho }^{\mathcal{H}}\le \left\{\begin{array}{cc}{\delta }_C^{\mathcal{H}}(r-1)+{\displaystyle \frac{c+3}{4}}+{\displaystyle \frac{3(c-1)}{4r}}{cos}^2\theta -{\displaystyle \frac{c-1}{2r}},\hfill & {\xi }_2\in \Gamma (Ran{F}_{\ast }),\hfill \\ {\delta }_C^{\mathcal{H}}(r-1)+{\displaystyle \frac{c+3}{4}}+{\displaystyle \frac{3(c-1)}{4(r-1)}}{cos}^2\theta ,\hfill & {\xi }_2\in \Gamma \left({(Ran{F}_{\ast })}^{\perp }\right),\hfill \end{array}\right.$$

and

$${\rho }^{\mathcal{H}}\le \left\{\begin{array}{cc}{\widehat{\delta }}_C^{\mathcal{H}}(r-1)+{\displaystyle \frac{c+3}{4}}+{\displaystyle \frac{3(c-1)}{4r}}{cos}^2\theta -{\displaystyle \frac{c-1}{2r}},\hfill & {\xi }_2\in \Gamma (Ran{F}_{\ast }),\hfill \\ {\widehat{\delta }}_C^{\mathcal{H}}(r-1)+{\displaystyle \frac{c+3}{4}}+{\displaystyle \frac{3(c-1)}{4(r-1)}}{cos}^2\theta ,\hfill & {\xi }_2\in \Gamma \left({(Ran{F}_{\ast })}^{\perp }\right).\hfill \end{array}\right.$$

(2) 

If $M_2\left(c\right)$ is a Kenmotsu space form ($c_1={\displaystyle \frac{c-3}{4}},c_2=c_3={\displaystyle \frac{c+1}{4}}$), then

$${\rho }^{\mathcal{H}}\le \left\{\begin{array}{cc}{\delta }_C^{\mathcal{H}}(r-1)+{\displaystyle \frac{c-3}{4}}+{\displaystyle \frac{3(c+1)}{4r}}{cos}^2\theta -{\displaystyle \frac{c+1}{2r}},\hfill & {\xi }_2\in \Gamma (Ran{F}_{\ast }),\hfill \\ {\delta }_C^{\mathcal{H}}(r-1)+{\displaystyle \frac{c-3}{4}}+{\displaystyle \frac{3(c+1)}{4(r-1)}}{cos}^2\theta ,\hfill & {\xi }_2\in \Gamma \left({(Ran{F}_{\ast })}^{\perp }\right),\hfill \end{array}\right.$$

and similarly for the ${\widehat{\delta }}_C^{\mathcal{H}}$.

(3) 

If $M_2\left(c\right)$ is a cosymplectic space form ($c_1=c_2=c_3={\displaystyle \frac{c}{4}}$), then

$${\rho }^{\mathcal{H}}\le \left\{\begin{array}{cc}{\delta }_C^{\mathcal{H}}(r-1)+{\displaystyle \frac{c}{4}}+{\displaystyle \frac{3c}{4r}}{cos}^2\theta -{\displaystyle \frac{c}{2r}},\hfill & {\xi }_2\in \Gamma (Ran{F}_{\ast }),\hfill \\ {\delta }_C^{\mathcal{H}}(r-1)+{\displaystyle \frac{c}{4}}+{\displaystyle \frac{3c}{4(r-1)}}{cos}^2\theta ,\hfill & {\xi }_2\in \Gamma \left({(Ran{F}_{\ast })}^{\perp }\right),\hfill \end{array}\right.$$

and similarly for the ${\widehat{\delta }}_C^{\mathcal{H}}$.

(4) 

If $M_2\left(c\right)$ is an almost $C\left(\alpha \right)$ space form ($c_1={\displaystyle \frac{c+3{\alpha }^2}{4}},c_2=c_3={\displaystyle \frac{c-{\alpha }^2}{4}}$), then

$${\rho }^{\mathcal{H}}\le \left\{\begin{array}{cc}{\delta }_C^{\mathcal{H}}(r-1)+{\displaystyle \frac{c+3{\alpha }^2}{4}}+{\displaystyle \frac{3(c-{\alpha }^2)}{4r}}{cos}^2\theta -{\displaystyle \frac{c-{\alpha }^2}{2r}},\hfill & {\xi }_2\in \Gamma (Ran{F}_{\ast }),\hfill \\ {\delta }_C^{\mathcal{H}}(r-1)+{\displaystyle \frac{c+3{\alpha }^2}{4}}+{\displaystyle \frac{3(c-{\alpha }^2)}{4(r-1)}}{cos}^2\theta ,\hfill & {\xi }_2\in \Gamma \left({(Ran{F}_{\ast })}^{\perp }\right),\hfill \end{array}\right.$$

and similarly for the ${\widehat{\delta }}_C^{\mathcal{H}}$.

For generalized Sasakian space forms, simultaneous equality in both inequalities of Theorem 3 again characterizes total geodesicity, regardless of the position of ${\xi }_2$ relative to $Ran{F}_{\ast }$.

Corollary 4.

Let $F:(M_1^{m_1},g_1)\to (M_2^{m_2}(c_1,c_2,c_3),{\varphi }_2,{\xi }_2,{\eta }_2,g_2)$ be a pointwise slant Riemannian map with rank $r\ge 3$ and slant function θ. Then F is totally geodesic if and only if equality holds simultaneously in both inequalities of (43).

Proof. 

$(\Rightarrow )$ If $\nabla F_{\ast }\equiv 0$, then $h_{ij}^{\alpha }=0$ for all $i,j,\alpha$ by (23), whence ${\parallel h\parallel }^2=0$, $C^{\mathcal{H}}=0$, and ${\delta }_C^{\mathcal{H}}(r-1)={\widehat{\delta }}_C^{\mathcal{H}}(r-1)=0$ by (27) and (28). Both inequalities of (43) reduce to equalities, independently of $c_3$ and of the position of ${\xi }_2$.

$(\Leftarrow )$ Assume equality in both inequalities of (43). Within either case ${\xi }_2\in \Gamma (Ran{F}_{\ast })$ or ${\xi }_2\in \Gamma \left({(Ran{F}_{\ast })}^{\perp }\right)$, the two bounds share the same ambient terms $c_1$, ${\displaystyle \frac{3c_2}{r}}{cos}^2\theta$, and $-{\displaystyle \frac{2}{r}}{c}_3$; subtracting the two equalities cancels these and leaves

$${\displaystyle \frac{r+1}{2r}}infC\left(L\right)+{\displaystyle \frac{2r-1}{2r}}supC\left(L\right)={\displaystyle \frac{3}{2}}{C}^{\mathcal{H}}.$$

(44)

By the equality conditions of Theorem 3, there exist orthonormal bases in which $h_{ij}^{\alpha }=0$ $(i\ne j)$, $h_{11}^{\alpha }=\cdots =h_{r-1,r-1}^{\alpha }=a^{\alpha }$, and $h_{rr}^{\alpha }=2a^{\alpha }$. Put $S:={\sum }_{\alpha }{\left(a^{\alpha }\right)}^2$. Then $C^{\mathcal{H}}={\displaystyle \frac{r+3}{r}}S$, while for a coordinate hyperplane retaining $Z_r$, $C\left(L\right)={\displaystyle \frac{1}{r-1}}{\sum }_{\alpha }[(r-2)+4]{\left(a^{\alpha }\right)}^2={\displaystyle \frac{r+2}{r-1}}S>S$. Since the second fundamental form is diagonal in the chosen basis, these coordinate hyperplanes realize the extreme Casorati values, so $infC\left(L\right)=S$ and $supC\left(L\right)={\displaystyle \frac{r+2}{r-1}}S$. Substituting into (44) and clearing $2r(r-1)$ gives

$${\displaystyle \frac{3(r-2)}{2r(r-1)}}S=0.$$

As $r\ge 3$, this forces $S=0$; hence every $a^{\alpha }=0$ and $\nabla F_{\ast }\equiv 0$, so F is totally geodesic. □

Remark 8.

When $\theta =0$, these reduce to the invariant inequalities for Sasakian space forms in Polat et al. [29]. When $\theta =\pi /2$, the ${cos}^2\theta$ term vanishes and we obtain inequalities involving only $c_1$ and $c_3$ if ${\xi }_2\in Ran{F}_{\ast }$, consistent with the anti-invariant results in [29,30].

5. Examples

We provide explicit examples illustrating the main theorems and the sharpness of our inequalities.

Example 1.

Following the construction of slant submanifolds in complex Euclidean spaces due to Chen–Tazawa [49] and Tazawa [50] (see also [51]), we construct a proper pointwise slant Riemannian map into a Kähler manifold and realize the equality case of Theorem 2.

Let $(M_2,J,g_2)=({\mathbb{C}}^4,J,g_{\mathrm{std}})$ with coordinates $(u_1,v_1,u_2,v_2,u_3,v_3,u_4,v_4)$ and $J(\partial /\partial u_i)=\partial /\partial v_i$, $J(\partial /\partial v_i)=-\partial /\partial u_i$ for $i=1,2,3,4$. Let $M_1={\mathbb{R}}^6$ with coordinates $(a_1,b_1,a_2,b_2,t_1,t_2)$ and the standard flat metric. Fix $\theta \in (0,\pi /2)$ and define $F:{\mathbb{R}}^6\to {\mathbb{C}}^4$ by

$$F(a_1,b_1,a_2,b_2,t_1,t_2)=\left(a_1,b_{1}cos\theta ,a_2,b_{2}cos\theta ,b_{1}sin\theta ,0,b_{2}sin\theta ,0\right).$$

(45)

Since F is independent of $(t_1,t_2)$,

$$ker{F}_{\ast }=span\left\{\partial /\partial t_1,\partial /\partial t_2\right\}\ne \left\{0\right\},$$

On ${(ker{F}_{\ast })}^{\perp }$, the differential gives

$$\begin{array}{cccc}\hfill E_1& ={\displaystyle \frac{\partial }{\partial u_1}},\hfill & \hfill E_2& =cos\theta {\displaystyle \frac{\partial }{\partial v_1}}+sin\theta {\displaystyle \frac{\partial }{\partial u_3}},\hfill \\ \hfill E_3& ={\displaystyle \frac{\partial }{\partial u_2}},\hfill & \hfill E_4& =cos\theta {\displaystyle \frac{\partial }{\partial v_2}}+sin\theta {\displaystyle \frac{\partial }{\partial u_4}},\hfill \end{array}$$

with $g_2(E_i,E_j)={\delta }_{ij}$, so F is a Riemannian map of rank $r=4$. Since $dimRan{F}_{\ast }=4<8$, we have ${(Ran{F}_{\ast })}^{\perp }\ne \left\{0\right\}$. Hence, F is a proper Riemannian map. The normal space is

$${(Ran{F}_{\ast })}^{\perp }=span\left\{{\nu }_1,{\nu }_2,{\displaystyle \frac{\partial }{\partial v_3}},{\displaystyle \frac{\partial }{\partial v_4}}\right\},$$

where ${\nu }_1=-sin\theta \partial /\partial v_1+cos\theta \partial /\partial u_3$ and ${\nu }_2=-sin\theta \partial /\partial v_2+cos\theta \partial /\partial u_4$.

We verify the slant condition on each basis vector

E₁:

$J\left(E_1\right)=\partial /\partial v_1=cos\theta E_2-sin\theta {\nu }_1$, so ${\left(J{E}_1\right)}^{Ran}=cos\theta E_2$ and $\parallel {\left(J{E}_1\right)}^{Ran}\parallel =cos\theta$.

E₂:

$J\left(E_2\right)=-cos\theta \partial /\partial u_1+sin\theta \partial /\partial v_3$; since $\partial /\partial v_3\perp E_i$ for all i, ${\left(J{E}_2\right)}^{Ran}=-cos\theta E_1$ and $\parallel {\left(J{E}_2\right)}^{Ran}\parallel =cos\theta$.

E₃:

$J\left(E_3\right)=\partial /\partial v_2=cos\theta E_4-sin\theta {\nu }_2$, so ${\left(J{E}_3\right)}^{Ran}=cos\theta E_4$ and $\parallel {\left(J{E}_3\right)}^{Ran}\parallel =cos\theta$.

E₄:

$J\left(E_4\right)=-cos\theta \partial /\partial u_2+sin\theta \partial /\partial v_4$; since $\partial /\partial v_4\perp E_i$ for all i, ${\left(J{E}_4\right)}^{Ran}=-cos\theta E_3$ and $\parallel {\left(J{E}_4\right)}^{Ran}\parallel =cos\theta$.

Hence, F is a properly pointwise slant Riemannian map with slant angle $\theta \in (0,\pi /2)$, ${\left(P^{\mathcal{R}}\right)}^2=-{cos}^2\theta ·I$, and ${\parallel P^{\mathcal{R}}\parallel }^2=4{cos}^2\theta =r{cos}^2\theta$, confirming (15).

Since ${\mathbb{C}}^4$ is flat ($c_1=c_2=0$), Theorem 2 gives

$${\rho }^{\mathcal{H}}\le {\delta }_C^{\mathcal{H}}(r-1)+c_1+{\displaystyle \frac{3c_2}{r-1}}{cos}^2\theta ={\delta }_C^{\mathcal{H}}\left(3\right).$$

(46)

Since F is affine, $\nabla F_{\ast }\equiv 0$, so $C^{\mathcal{H}}=0$, ${\delta }_C^{\mathcal{H}}\left(3\right)=0$, and ${\rho }^{\mathcal{H}}=0$. Inequality (46) holds with equality for every $\theta \in (0,\pi /2)$, realizing the quasi-umbilical equality case (Definition 8) with $h_{ij}^{\alpha }\equiv 0$.

Example 2.

We construct a proper Riemannian map that is properly pointwise slant with a non-constant slant function, confirming Proposition 2.

Let $M_1=\{(u,v,t)\in {\mathbb{R}}^3:v\ne 0\}$ carry the metric $g_1=(1+v^2)d{u}^2+d{v}^2+d{t}^2$, and let $({\mathbb{C}}^2,J,g_{\mathrm{std}})$ have coordinates $(u_1,v_1,u_2,v_2)$ with $J(\partial /\partial u_i)=\partial /\partial v_i$, $J(\partial /\partial v_i)=-\partial /\partial u_i$.

Define $F:M_1\to {\mathbb{C}}^2$ by

$$F(u,v,t)=\left(u,0,vcosu,vsinu\right).$$

Since $ker{F}_{\ast }=span\{\partial /\partial t\}\ne \left\{0\right\}$, F is not an immersion. On ${(ker{F}_{\ast })}^{\perp }$,

$$F_{\ast }\left({\displaystyle \frac{\partial }{\partial u}}\right)={\displaystyle \frac{\partial }{\partial u_1}}-vsinu{\displaystyle \frac{\partial }{\partial u_2}}+vcosu{\displaystyle \frac{\partial }{\partial v_2}},F_{\ast }\left({\displaystyle \frac{\partial }{\partial v}}\right)=cosu{\displaystyle \frac{\partial }{\partial u_2}}+sinu{\displaystyle \frac{\partial }{\partial v_2}}.$$

A direct computation gives

$$\parallel F_{\ast }{\partial }_u{\parallel }^2=1+v^2,{\parallel F_{\ast }{\partial }_v\parallel }^2=1,and{g}_2(F_{\ast }{\partial }_u,F_{\ast }{\partial }_v)=0,$$

matching the coefficients of $g_1$, so F is a Riemannian map of rank $r=2$. Since $dimRan{F}_{\ast }=2<4$, ${(Ran{F}_{\ast })}^{\perp }\ne \left\{0\right\}$, therefore F is a proper Riemannian map.

Set $e_1={(1+v^2)}^{-1/2}{F}_{\ast }{\partial }_u$ and $e_2=F_{\ast }{\partial }_v$, an orthonormal basis of $Ran{F}_{\ast }$. Computing the tangential part $P^{\mathcal{R}}$ of J gives

$$P^{\mathcal{R}}{e}_1=-{\displaystyle \frac{v}{\sqrt{1+v^2}}}{e}_2,P^{\mathcal{R}}{e}_2={\displaystyle \frac{v}{\sqrt{1+v^2}}}{e}_1,{\left(P^{\mathcal{R}}\right)}^2=-{\displaystyle \frac{v^2}{1+v^2}}I.$$

Both basis directions yield the same ratio, so F is properly pointwise slant with non-constant slant function

$$cos\theta \left(v\right)={\displaystyle \frac{\left|v\right|}{\sqrt{1+v^2}}}\in (0,1),$$

which varies with position, $\theta \to \pi /2$ as $v\to 0^{+}$ and $\theta \to 0$ as $v\to +\infty$, and the norm identity

$$\parallel P^{\mathcal{R}}{\parallel }^2={\displaystyle \frac{2v^2}{1+v^2}}=r{cos}^2\theta$$

holds, confirming Proposition 2. This example thus illustrates a pointwise slant Riemannian map with a non-constant slant function and verifies the norm identity of Proposition 2 for variable angle. We note that Proposition 2 is an algebraic identity valid for every rank, so this rank-2 map serves purely as an illustration of it and is not governed by Theorems 2 and 3. Indeed, by Remark 2, a proper pointwise slant map into an almost Hermitian manifold requires $r\ge 4$.

Example 3.

We use Corollary 3 to derive a sharp non-existence obstruction for properly slant proper Riemannian maps into the standard Sasakian space form.

Let $(M_2,{\varphi }_2,{\xi }_2,{\eta }_2,g_2)=({\mathbb{R}}^9,{\varphi }_2,{\xi }_2,{\eta }_2,g_{\mathrm{std}})$ carry the standard Sasakian structure of constant ${\varphi }_2$-sectional curvature $c=-3$, so that

$$c_1={\displaystyle \frac{c+3}{4}}=0,c_2=c_3={\displaystyle \frac{c-1}{4}}=-1.$$

Suppose, for contradiction, that there exists a pointwise slant Riemannian map $F:({\mathbb{R}}^6,g_{\mathrm{std}})\to ({\mathbb{R}}^9,{\varphi }_2,{\xi }_2,{\eta }_2,g_{\mathrm{std}})$ of rank $r=4$ with $ker{F}_{\ast }=span\{\partial /\partial t_1,\partial /\partial t_2\}$ (so $ker{F}_{\ast }\ne \left\{0\right\}$ and, since $dimRan{F}_{\ast }=4<9$, ${(Ran{F}_{\ast })}^{\perp }\ne \left\{0\right\}$), slant angle $\theta \in (0,\pi /2)$, vanishing second fundamental form $\nabla F_{\ast }\equiv 0$, and ${\xi }_2\in \Gamma \left({(Ran{F}_{\ast })}^{\perp }\right)$.

Since $\nabla F_{\ast }\equiv 0$, we have $C^{\mathcal{H}}=0$ and ${\delta }_C^{\mathcal{H}}\left(3\right)=0$. Since ${\mathbb{R}}^6$ is flat, ${\rho }^{\mathcal{H}}=0$. Applying Corollary 3 with ${\xi }_2\in \Gamma \left({(Ran{F}_{\ast })}^{\perp }\right)$

$$0={\rho }^{\mathcal{H}}\le {\delta }_C^{\mathcal{H}}\left(3\right)+{\displaystyle \frac{c+3}{4}}+{\displaystyle \frac{3(c-1)}{4(r-1)}}{cos}^2\theta =0+0+{\displaystyle \frac{3(-4)}{4·3}}{cos}^2\theta =-{cos}^2\theta .$$

(47)

This gives $0\le -{cos}^2\theta$, which is impossible for $\theta \in (0,\pi /2)$.

Thus, no proper totally geodesic slant Riemannian map $F:{\mathbb{R}}^6\to {\mathbb{R}}^9$ with $\nabla F_{\ast }\equiv 0$, ${\xi }_2\in \Gamma \left({(Ran{F}_{\ast })}^{\perp }\right)$, and $\theta \in [0,\pi /2)$ can exist. The unique compatible case is $\theta =\pi /2$, where (47) reduces to $0\le 0$.

Remark 9.

The same obstruction holds for any Sasakian space form with $c<1$ (so that $c_2<0$) and ${\xi }_2\in \Gamma \left({(Ran{F}_{\ast })}^{\perp }\right)$: no totally geodesic slant Riemannian map from a flat source manifold can exist. This gives a family of sharp non-existence results parametrized by c. Consistent with Remark 8: when $\theta =0$ the bound recovers the invariant case of Polat et al. [29], and when $\theta =\pi /2$ the ${cos}^2\theta$ term vanishes, giving $0\le 0$ as the only compatible case.

Example 4.

We use Corollary 3 to establish a non-existence obstruction for pointwise slant proper Riemannian maps into a Kenmotsu space form, and show that this obstruction is strictly stronger than that of Example 3.

Let $(M_2,{\varphi }_2,{\xi }_2,{\eta }_2,g_2)=(M^9\left(c\right),{\varphi }_2,{\xi }_2,{\eta }_2,g_{\mathrm{std}})$ be a Kenmotsu space form with constant ${\varphi }_2$-sectional curvature $c=-3$, chosen here for concreteness (the general dependence on c is discussed in Remark 10), so that

$$c_1={\displaystyle \frac{c-3}{4}}=-{\displaystyle \frac{3}{2}},c_2=c_3={\displaystyle \frac{c+1}{4}}=-{\displaystyle \frac{1}{2}}.$$

Suppose, for contradiction, that there exists a pointwise slant Riemannian map $F:({\mathbb{R}}^6,g_{\mathrm{std}})\to (M^9(-3),{\varphi }_2,{\xi }_2,{\eta }_2,g_{\mathrm{std}})$ of rank $r=4$ with $ker{F}_{\ast }=span\{\partial /\partial t_1,\partial /\partial t_2\}$ (so $ker{F}_{\ast }\ne \left\{0\right\}$ and, since $dimRan{F}_{\ast }=4<9$, ${(Ran{F}_{\ast })}^{\perp }\ne \left\{0\right\}$), constant slant angle $\theta \in (0,\pi /2)$, vanishing second fundamental form $\nabla F_{\ast }\equiv 0$, and ${\xi }_2\in \Gamma \left({(Ran{F}_{\ast })}^{\perp }\right)$.

Since $\nabla F_{\ast }\equiv 0$, by (23) we have $h_{ij}^{\alpha }=0$ for all $i,j,\alpha$. Consequently, ${\parallel h\parallel }^2=0$ by (24), so $C^{\mathcal{H}}=0$ by (25), and it follows from (27) and (28) that ${\delta }_C^{\mathcal{H}}\left(3\right)=0$ and ${\widehat{\delta }}_C^{\mathcal{H}}\left(3\right)=0$. Since ${\mathbb{R}}^6$ is flat, ${\rho }^{\mathcal{H}}=0$.

Applying Corollary 3 (2) with ${\xi }_2\in \Gamma \left({(Ran{F}_{\ast })}^{\perp }\right)$

$$0={\rho }^{\mathcal{H}}\le {\delta }_C^{\mathcal{H}}\left(3\right)+{\displaystyle \frac{c-3}{4}}+{\displaystyle \frac{3(c+1)}{4(r-1)}}{cos}^2\theta =0+\left(-{\displaystyle \frac{3}{2}}\right)+{\displaystyle \frac{3\left(-\frac{1}{2}\right)}{3}}{cos}^2\theta =-{\displaystyle \frac{3}{2}}-{\displaystyle \frac{1}{2}}{cos}^2\theta .$$

(48)

Since $\theta \in (0,\pi /2)$, we have ${cos}^2\theta \in (0,1)$, so

$$-{\displaystyle \frac{3}{2}}-{\displaystyle \frac{1}{2}}{cos}^2\theta <-{\displaystyle \frac{3}{2}}<0.$$

This gives $0\le -{\displaystyle \frac{3}{2}}-{\displaystyle \frac{1}{2}}{cos}^2\theta <0$, a contradiction.

Therefore, no totally geodesic pointwise slant Riemannian map $F:{\mathbb{R}}^6\to M^9(-3)$ with $\nabla F_{\ast }\equiv 0$, ${\xi }_2\in \Gamma {(Ran{F}_{\ast })}^{\perp }$, and slant angle $\theta \in (0,\pi /2)$ can exist. Unlike Example 3, where the anti-invariant angle $\theta =\pi /2$ remained compatible, here the obstruction is total; the term $c_1=-{\displaystyle \frac{3}{2}}<0$ is independent of θ and contributes a strictly negative constant regardless of the slant angle. Consequently, even substituting $\theta =\pi /2$ into (48) yields $0\le -{\displaystyle \frac{3}{2}}<0$, which is still a contradiction. No slant angle $\theta \in [0,\pi /2]$ is compatible with the existence of such a map. The specific value $c=-3$ is not essential: as shown in Remark 10, the same total obstruction holds for every Kenmotsu space form with $c<3$, so that $c_1<0$.

Remark 10.

The contrast between Examples 3 and 4 reflects a fundamental difference between Sasakian and Kenmotsu space forms with the same value $c=-3$. In the Sasakian case, $c_1={\displaystyle \frac{c+3}{4}}=0$, so the bound in Corollary 3 (1) reduces to ${\displaystyle \frac{3c_2}{r-1}}{cos}^2\theta =-{cos}^2\theta$, which vanishes at $\theta =\pi /2$ and leaves the anti-invariant case compatible. In the Kenmotsu case, $c_1={\displaystyle \frac{c-3}{4}}=-{\displaystyle \frac{3}{2}}<0$ regardless of θ, so the bound $c_1+{\displaystyle \frac{3c_2}{r-1}}{cos}^2\theta =-{\displaystyle \frac{3}{2}}-{\displaystyle \frac{1}{2}}{cos}^2\theta$ is strictly negative for every $\theta \in [0,\pi /2]$. The obstruction is therefore driven by $c_1$ in the Kenmotsu case and by $c_2{cos}^2\theta$ in the Sasakian case. More generally, the same total obstruction holds for any Kenmotsu space form with $c<3$ (so that $c_1<0$), yielding a family of sharp non-existence results parametrized by c.

6. Conclusions and Open Problems

The central technical contribution of this paper is the norm identity ${\parallel P\parallel }^2=d{cos}^2\theta$ (Proposition 2), which allows the range-space scalar curvature ${\rho }^{\mathcal{R}}$ to be expressed as an explicit function of the slant angle $\theta$ via the Gauss equation. This identity is the bridge between the slant geometry of the map and the algebraic optimization argument of Lemma 1, and it is what makes the inequalities sharp rather than merely valid. The factor ${cos}^2\theta$ is not an artifact of the proof but carries genuine geometric meaning: it measures how much of the almost complex or almost contact structure is captured by the range distribution at each point, interpolating between full capture ($\theta =0$, invariant) and full escape ($\theta =\pi /2$, anti-invariant). Building on this identity, we have established optimal Casorati inequalities involving the normalized curvatures ${\delta }_C^{\mathcal{H}}$ and ${\widehat{\delta }}_C^{\mathcal{H}}$ for pointwise slant Riemannian maps into generalized complex space forms (Theorem 2) and generalized Sasakian space forms (Theorem 3), the latter treating separately the cases ${\xi }_2\in \Gamma (Ran{F}_{\ast })$ and ${\xi }_2\in \Gamma \left({(Ran{F}_{\ast })}^{\perp }\right)$. These subsume, as limiting cases $\theta =0$ and $\theta =\pi /2$, all previously known results for invariant and anti-invariant Riemannian maps [28,29]. Specializing to Sasakian, Kenmotsu, cosymplectic, and almost $C\left(\alpha \right)$ space forms yields explicit inequalities (Corollary 3). A consequence worth noting is that the equality case is the same quasi-umbilical condition $h_{11}^{\alpha }=\cdots =h_{r-1,r-1}^{\alpha }={\displaystyle \frac{1}{2}}{h}_{rr}^{\alpha }$ regardless of $\theta$: the slant angle controls the level of the bound but not the shape of the extremal configuration. Moreover, Corollaries 2 and 4 show that simultaneous equality in both the ${\delta }_C^{\mathcal{H}}$ and ${\widehat{\delta }}_C^{\mathcal{H}}$ inequalities characterizes totally geodesic maps, providing a complete geometric description of the equality case that is independent of both the slant angle $\theta$ and the ambient curvature parameters. As applications, Examples 3 and 4 derive sharp non-existence obstructions for totally geodesic pointwise slant proper Riemannian maps into Sasakian and Kenmotsu space forms respectively, revealing a structural contrast: in the Sasakian case, the anti-invariant boundary $\theta =\pi /2$ remains compatible, while in the Kenmotsu case, the negative value of $c_1$ rules out all slant angles simultaneously.

We propose the following open problems for future work.

(1) 

Casorati inequalities for semi-slant, bi-slant, and hemi-slant Riemannian maps to generalized complex and Sasakian space forms remain open. Partial results for bi-slant maps to Kenmotsu manifolds appear in [30].

(2) 

Casorati inequalities for pointwise slant Riemannian maps to nearly Kähler or nearly Sasakian targets remain unexplored; the invariant case was treated in [31].

(3) 

Casorati inequalities for pointwise slant Riemannian maps between statistical manifolds would connect the present results to information geometry via the interplay between $\theta$ and the difference tensor of the statistical structure.