Geometry of Riemannian Maps from Generic Submanifolds of Kähler Manifolds

Abstract

This paper extends the theory of Riemannian maps to the setting of generic submanifolds of Kähler manifolds. We introduce the notion of holomorphic Riemannian maps from generic submanifolds and establish fundamental relations between the geometric structures involved. Our main results include a characterization of when the image distribution inherits a Kähler structure, a harmonicity criterion for such maps, and a relation between holomorphic sectional curvatures. The theory developed here generalizes previous work on CR-submanifolds while demonstrating new phenomena specific to the generic case. Several explicit examples illustrate the non-trivial nature of our results.

1. Introduction

The study of submanifolds of almost Hermitian manifolds constitutes a rich chapter in differential geometry, with important classes including holomorphic, totally real, CR-, slant, and generic submanifolds. Among these, CR-submanifolds introduced by Bejancu [1] have been especially influential, unifying the study of both holomorphic and totally real submanifolds within a single geometric setting. The geometry of CR-submanifolds of Kähler manifolds has been extensively investigated by Chen [2,3] and many others.

Parallel to submanifold theory, the theory of Riemannian submersions initiated by O’Neill [4] has developed into an important area of research, with significant applications in Riemannian geometry and mathematical physics. A particularly interesting synthesis of these two research directions was achieved by Kobayashi [5], who initiated the study of Riemannian submersions from CR-submanifolds of Kähler manifolds. This study has been further extended by Deshmukh et al. [6,7], in which they obtained the relations between the Ricci curvatures and the scalar curvatures of a Kähler manifold and the base manifold. In [8,9], S. Ali et al. studied the effects of such submersions when the ambient manifold is nearly Kähler. This line of research was later extended by Fatima and Ali [10,11] into the broader setting of generic submanifolds of Kähler manifolds, a class introduced by Chen [2], in which the complementary distribution is purely real rather than totally real. Further contributions in this direction have been made by Fatima et al. [12] and Şahin et al. [13]. For more literature on generic submersion one can go through [14,15].

Fischer [16] later introduced Riemannian maps as a generalization unifying both Riemannian submersions and isometric immersions. This broader perspective has led to numerous developments, including the work of Şahin [17,18,19,20,21] on various classes of Riemannian maps between almost Hermitian manifolds. Recent works have studied Riemannian maps on Ricci solitons [22,23], warped products [24], Clairaut conditions [25,26,27] and helices [28], showing that the theory continues to expand rapidly. Most recently, this research thread has evolved to examine Riemannian maps from CR-submanifolds of Kähler manifolds to almost Hermitian manifolds [29], building directly on the earlier work of Kobayashi, Deshmukh, and Ali.

In this paper, we introduce Riemannian maps from generic submanifolds of Kähler manifolds to an almost Hermitian manifold and establish

• a necessary and sufficient condition for the image of a holomorphic Riemannian map to be Kähler (Theorem 1);
• a harmonicity criterion in terms of the mean curvature of the purely real distribution (Theorem 2);
• a holomorphic sectional curvature relation exhibiting an extra correction term that vanishes precisely when the submanifold is mixed-geodesic (Theorem 3);
• a complete Ricci tensor identity that systematically collects all contributions from the generic structure (Theorem 4);
• a scalar curvature identity (Theorem 5).

Our work shows that the generic case presents distinctive features not present in the CR setting, necessitating new technical tools and yielding richer geometric structure. The presence of the O’Neill-type tensor C for the generic structure introduces additional terms in all fundamental relations, making our results proper generalizations of the CR case.

After establishing the necessary preliminaries in Section 2, we prove in Section 3 our main lemma establishing fundamental relations between the second fundamental form of the submanifold and the O’Neill tensors of the Riemannian map. This section also provides a characterization of when the image distribution inherits a Kähler structure, while Section 4 provides a harmonicity criterion for such maps, establishes a relation between holomorphic sectional curvatures, and obtains complete Ricci and scalar curvature identities. Section 5 presents detailed examples verifying our results. Our work demonstrates that Riemannian maps from generic submanifolds provide a fertile ground for further research in Hermitian geometry.

2. Preliminaries

To make this paper comprehensive, we first collect the minimal background on Kähler manifolds, generic submanifolds, and Riemannian maps.

Let $\overline{M}$ be a Kähler manifold of complex dimension m, equipped with a Kähler metric g. It should be noted that in this case, $\overline{M}$ has a complex structure J satisfying the properties given by

$$J^2=-I,g({\zeta }_1,{\zeta }_2)=g(J{\zeta }_1,J{\zeta }_2),\left({\overline{\nabla }}_{{\zeta }_1}J\right){\zeta }_2=0,$$

for each vector field ${\zeta }_1,{\zeta }_2\in \mathrm{\Gamma }\left(T\overline{M}\right)$, where $\overline{\nabla }$ is the Levi-Civita connection on $\overline{M}$. Let $\overline{R}$ be the Riemannian curvature tensor field of $\overline{M}$. Consider a unit vector U at a point p of $\overline{M}$. Then, the pair $\{U,JU\}$ determines a plane $\pi$ called a holomorphic section whose curvature $K^{\overline{M}}$ is given by

$$K^{\overline{M}}=g(\overline{R}(U,JU)JU,U),$$

and is called the holomorphic sectional curvature with respect to U [30].

Let M be a real submanifold of $\overline{M}$ of real dimension m. Hence, the Riemannian metric induced on M by the Kählerian metric g is denoted by the same symbol g. Then, Gauss and Weingarten formulae are given by

$${\overline{\nabla }}_{{\zeta }_1}{\zeta }_2={\nabla }_{{\zeta }_1}^N{\zeta }_2+h({\zeta }_1,{\zeta }_2),$$

(1)

$${\overline{\nabla }}_{{\zeta }_1}\xi =-{\mathcal{S}}_{\xi }{\zeta }_1+{\nabla }_{{\zeta }_1}^{\perp }\xi ,$$

(2)

for any vector fields ${\zeta }_1,{\zeta }_2\in \mathrm{\Gamma }\left(TM\right)$ and $\xi \in$ $\mathrm{\Gamma }\left(T^{\perp }M\right)$, where represents the second fundamental form, ${\nabla }^{\perp }$ is the linear connection in the normal bundle $T^{\perp }M$ and ${\mathcal{S}}_{\zeta }$ is the Weingarten map in the direction of $\zeta$. Let R^M and $\overline{R}$ be the curvature tensor fields of M and $\overline{M}$ respectively. Then, Gauss and Weingarten formulae imply

$$\begin{array}{c}\hfill g(\overline{R}({\zeta }_1,{\zeta }_2){\zeta }_3,{\zeta }_4)=g(R^M({\zeta }_1,{\zeta }_2){\zeta }_3,{\zeta }_4)-g(h({\zeta }_2,{\zeta }_3),h({\zeta }_1,{\zeta }_4\left)\right)+g(h({\zeta }_1,{\zeta }_3),h({\zeta }_2,{\zeta }_4\left)\right)\end{array}$$

(3)

for ${\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4\in \mathrm{\Gamma }(TM)$.

Now, suppose M is a real submanifold of an almost Hermitian manifold $\overline{M}$ with almost complex structure J. Let $D_p=T_{p}M\cap J{T}_{p}M$, $p\in M$ be the maximal complex subspace of the tangent space $T_{p}M$. If the dimension of $D_p$ is constant at each point $p\in M$, and it defines a differentiable distribution on M, then M is called a generic submanifold of $\overline{M}$ [2].

We call D the holomorphic distribution, and the orthogonal complementary distribution $D^{\perp }$ of D in $TM$ is the purely real distribution, which satisfy the following

$$D\perp D^{\perp },D^{\perp }\cap J{D}^{\perp }=\left\{0\right\}.$$

Let $\nu$ define a differentiable vector sub-bundle of $T^{\perp }M$ satisfying

$$T^{\perp }M=F{D}^{\perp }\oplus \nu ,t\left(T^{\perp }M\right)=D^{\perp },$$

where $F:TM\to T^{\perp }M$ is the normal component of J on TM and $t:T^{\perp }M\to TM$ is the tangential part of ${J|}_{T^{\perp }M}$. Let $P:TM\to TM$ be the tangential part of J on $TM$, so that $JX=PX+FX$ for any $X\in \mathrm{\Gamma }\left(TM\right)$. Then, for a generic submanifold M,

$$PD=D\mathrm{and}P{D}^{\perp }\subset D^{\perp }.$$

It is known that the horizontal distribution D of a generic submanifold M of a Kähler manifold $\overline{M}$ is integrable if and only if

$$g\left(h\right(X,JY),FZ)=g\left(h\right(Y,JX),FZ),$$

for any $X,Y\in D$ and $Z\in D^{\perp }$ and the vertical distribution $D^{\perp }$ is integrable if and only if

$${\nabla }_{Z}PW-{\nabla }_{W}PZ+{\tilde{A}}_{FZ}W-{\tilde{A}}_{FW}Z\in D^{\perp },$$

for any vector fields $Z,W\in D^{\perp }$.

Let $(M^m,g_M)$ and $(N^n,g_N)$ be two Riemannian manifolds, where $dim\left(M\right)=m$, $dim\left(N\right)=n$ and $m>n$. A Riemannian submersion $\pi :M⟶N$ is a map of M onto N satisfying the following axioms:

(1)

$\pi$ has maximal rank.

(2)

The differential ${\pi }_{\ast }$ preserves the lengths of horizontal vectors.

For each $q\in N$, ${\pi }^{-1}\left(q\right)$ is an $(m-n)$ dimensional submanifold of M. The submanifolds ${\pi }^{-1}\left(q\right)$, $q\in N$, are called fibers. A vector field on M is called vertical if it is always tangent to fibres. A vector field on M is called horizontal if it is always orthogonal to fibres. A vector field ${\zeta }_1$ on M is called basic if ${\zeta }_1$ is horizontal and $\pi$-related to a vector field ${{\zeta }_1}_{\ast }$ on N. Note that we denote the projection morphisms on the distributions $ker{\pi }_{\ast }$ and ${(ker{\pi }_{\ast })}^{\perp }$ by $\mathcal{V}$ and $\mathcal{H}$, respectively. For basic vector fields ${\xi }_1,{\xi }_2$ on M, we have

$${\nabla }_{{\xi }_1}^M{\xi }_2={\mathcal{A}}_{{\xi }_1}{\xi }_2+\mathcal{H}{\nabla }_{{\xi }_1}^M{\xi }_2,$$

where ${\nabla }^M$ is the Levi-Civita connection on M and $\mathcal{A}$ is the O’Neill tensor field, which is anti-symmetric on the set of horizontal vector fields [4].

We have the following lemma for basic vector fields [4].

Lemma 1.

Let X and Y be any basic vector fields on M. Then:

(i)

$g(X,Y)=g_N(X_{\ast },Y_{\ast })\circ \pi$.

(ii)

The horizontal part $\mathcal{H}[X,Y]$ of $[X,Y]$ is a basic vector field and corresponds to $[X_{\ast },Y_{\ast }]$; that is ${\pi }_{\ast }\mathcal{H}[X,Y]=[X_{\ast },Y_{\ast }]\circ \pi$.

(iii)

$[V,X]\in D^{\perp }$ for any $V\in D^{\perp }$.

(iv)

$\mathcal{H}\left({\nabla }_{X}Y\right)$ is a basic vector field corresponding to ${\nabla }_{X_{\ast }}^{\ast }{Y}_{\ast }$, where ${\nabla }^{\ast }$ is the Riemannian connection on N.

By observing the integrability of the anti-invariant distribution of a CR-submanifold and the vertical distribution of a Riemannian submersion, Kobayashi introduced the submersion CR-submanifolds of Kähler manifolds as follows [5].

Definition 1.

Let M be a $CR$-submanifold of an almost Hermitian manifold $(\overline{M},g,J)$ with distributions $\mathcal{D}$ and ${\mathcal{D}}^{\perp }$ and the normal bundle ν. By a submersion of M onto an almost Hermitian manifold B we mean a Riemannian submersion $\pi :M\to B$ along with the following conditions:

(i)

$D^{\perp }$ is the kernel of ${\pi }_{\ast }$; that is, ${\pi }_{\ast }\left(D^{\perp }\right)=\left\{0\right\}$.

(ii)

${\pi }_{\ast }{D}_p=T_{\pi \left(p\right)}B$ is complex isometry, where $p\in D$ and $T_{\pi \left(p\right)}B$ is the tangent space of B at $\pi \left(p\right)$.

(iii)

J interchanges $D^{\perp }$ and ν; that is, $J{D}^{\perp }=\nu$.

He showed that under this situation B is necessarily a Kähler manifold and obtained the relation between holomorphic sectional curvatures of M restricted to D and those of B.

Let $\overline{\nabla },\nabla$ and ${\nabla }^{\ast }$ denote Riemannian connections on $\overline{M},M$ and B respectively. For the connection ${\nabla }^{\ast }$ we define the corresponding connection ${\nabla }^{\ast }$ for basic vector fields on M by

$${\overline{\nabla }}_X^{\ast }Y=\mathcal{H}\left({\nabla }_{X}Y\right).$$

Then ${\overline{\nabla }}_X^{\ast }Y$ is a basic vector field, and by Lemma 1, we have

$${\pi }_{\ast }\left({\overline{\nabla }}_X^{\ast }Y\right)={\nabla }_{X_{\ast }}^{\ast }{Y}_{\ast }.$$

We define a tensor field C on M by

$${\nabla }_{X}Y={\overline{\nabla }}_X^{\ast }Y+C(X,Y)$$

(4)

for any $X,Y\in D$, where $C(X,Y)$ is the vertical part of ${\nabla }_{X}Y$; i.e., $\mathcal{V}\left({\nabla }_{X}Y\right)=C(X,Y)$. It has been observed that C is skew-symmetric and satisfies

$$C(X,Y)={\displaystyle \frac{1}{2}}\mathcal{V}[X,Y]$$

for any $X,Y\in D$. Also for $X\in D$ and $V\in D^{\perp }$, we define an operator $\mathcal{A}$ on M by

$${\nabla }_{X}V=\mathcal{V}\left({\nabla }_{X}V\right)+{\mathcal{A}}_{X}V,$$

where ${\mathcal{A}}_{X}V$ is the horizontal part of ${\nabla }_{X}V$. Since $[V,X]\in D^{\perp }$ for any basic vector field X and $V\in D^{\perp }$, we have

$$\mathcal{H}\left({\nabla }_{X}V\right)=\mathcal{H}\left({\nabla }_{V}X\right)={\mathcal{A}}_{X}V.$$

The operators C and $\mathcal{A}$ are related by

$$g({\mathcal{A}}_{X}V,Y)=-g(V,C(X,Y)),X,Y\in D\mathrm{and}V\in D^{\perp }.$$

The operator C in (4) was introduced by Kobayashi [5].

Generic submanifolds are a more general class of CR-submanifolds. Following the same approach, S. Ali and T. Fatima defined the submersion of generic submanifolds [10] as follows:

(i)

$D^{\perp }$ is the kernel of $\pi$; that is, ${\pi }_{\ast }\left(D^{\perp }\right)=\left\{0\right\}$,

(ii)

${\pi }_{\ast }\left(D_p\right)=T_{\pi \left(p\right)}B$ is a complex isometry, where $p\in M$ and $T_{\pi \left(p\right)}B$ is the tangent space of B at $\pi \left(p\right)$.

This concept remains an active research area. For interested readers, we refer to [11,12,13] etc.

Now, a smooth map $\pi :(M^m,g)⟶(B^n,g_B)$ is called a Riemannian map at $p_1\in M$ if the horizontal restriction

$${\pi }_{\ast p_1}^h:{(ker{\pi }_{\ast p_1})}^{\perp }⟶(range{\pi }_{\ast p_1})$$

is a linear isometry between the inner product spaces $({(ker{\pi }_{\ast p_1})}^{\perp },g(p_1){\mid }_{{(ker{\pi }_{\ast p_1})}^{\perp }})$ and $(range{\pi }_{\ast p_1},g_B\left(p_2\right){\mid }_{(range{\pi }_{\ast p_1})})$, $p_2=\pi \left(p_1\right)$ [16]. For a Riemannian map $\pi$, the second fundamental form $(\nabla {\pi }_{\ast })$ satisfies

$$(\nabla {\pi }_{\ast })({\zeta }_1,{\zeta }_2)\in \mathrm{\Gamma }\left({(ker{\pi }_{\ast })}^{\perp }\right),{\zeta }_1,{\zeta }_2\in \mathrm{\Gamma }\left({(ker{\pi }_{\ast })}^{\perp }\right),$$

as shown in [18]. Thus at $p\in M$, we write

$${{\nabla }^{\pi }}_{{\zeta }_1}{\pi }_{\ast }\left({\zeta }_2\right)\left(p\right)={\pi }_{\ast }\left({\nabla }_{{\zeta }_1}^M{\zeta }_2\right)\left(p\right)+(\nabla {\pi }_{\ast })({\zeta }_1,{\zeta }_2)\left(p\right),$$

where ${\nabla }^{\pi }$ is the pull-back connection. Hence we have

$$\begin{array}{cc}\hfill g_B(R^N({\pi }_{\ast }{\zeta }_1,{\pi }_{\ast }{\zeta }_2){\pi }_{\ast }{\zeta }_3,{\pi }_{\ast }{\zeta }_4)& =g(R^M({\zeta }_1,{\zeta }_2){\zeta }_3,{\zeta }_4)+g_B((\nabla {\pi }_{\ast })({\zeta }_1,{\zeta }_3),(\nabla {\pi }_{\ast })({\zeta }_2,{\zeta }_4))\hfill \\ \hfill & -g_B((\nabla {\pi }_{\ast })({\zeta }_2,{\zeta }_3),(\nabla {\pi }_{\ast })({\zeta }_1,{\zeta }_4))\hfill \end{array}$$

(5)

for ${\zeta }_3,{\zeta }_4\in \mathrm{\Gamma }{(ker{\pi }_{\ast })}^{\perp }$ [20].

A smooth map $\pi :(M,g)\to (B,g_B)$ between Riemannian manifolds is harmonic if it is a critical point of the energy functional. The Euler–Lagrange equation gives the tension field

$$\tau \left(\pi \right)=\mathrm{trace}(\nabla {\pi }_{\ast })=\sum _{i=1}^m(\nabla {\pi }_{\ast })(e_i,e_i),$$

where $\{e_1,\dots ,e_m\}$ is an orthonormal frame of $TM$. The map $\pi$ is harmonic if and only if $\tau \left(\pi \right)=0$ [31].

In [29], Sahin studied a holomorphic Riemannian map from a CR-submanifold and gave the following definition.

Definition 2.

Let M be a $CR$-submanifold of an almost Hermitian manifold ($\overline{M},\overline{j},\overline{g}$) with distributions D and $D^{\perp }$ and the normal bundle $T{M}^{\perp }$. By a Riemamian map of M to an almost Hermitian manifold ($B,J_B,g_B$) we mean a Riemannian map $\pi :M\to B$ with the following conditions:

• ${\mathcal{D}}^{\perp }$ is the kernel of ${\pi }_{\ast }$; that is, ${\pi }_{\ast }\left({\mathcal{D}}^{\perp }\right)=0$.
• ${\pi }_{\ast }$ is a holomorphic Riemannian map between $D_p$ and range ${\pi }_{\ast p}$ for $p\in M$, where (range ${\pi }_{\ast p}$) is the range of ${\pi }_{\ast }$ at $p\in M$.
• $\overline{J}$ interchanges ${\mathcal{D}}^{\perp }$ and $T{M}^{\perp }$; that is $\overline{J}\left({\mathcal{D}}^{\perp }\right)=T{M}^{\perp }$.

Recall that a holomorphic Riemannian map is

$$J_{B\pi \left(p\right)}{\pi }_{\ast p}={\pi }_{\ast p}\left({\overline{j}}_p\right)$$

for any $p\in M$.

Inspired by this new study, the present article studies the Riemannanian map from a generic submanifold of a Kähler manifold to an almost Hermitian manifold. During the study of Riemannian maps from generic submanifolds, the assumptions made in [10] are naturally considered to preserve the geometric structures.

Now, we proceed to derive the fundamental relations for Riemannian maps from generic submanifolds of a Kähler manifold to an almost Hermitian manifold.

3. Fundamental Relations for Riemannian Maps from Generic Submanifolds

In this section, we establish the fundamental technical results that underpin our entire theory. The following lemma reveals the intricate relationship between the second fundamental form of the submanifold immersion and the O’Neill tensors of the Riemannian map in the generic setting.

By utilizing the definition of submersion of a generic submanifold of a Kähler manifold in [10], we define the following.

Definition 3.

Let M be a generic submanifold of a Kähler manifold $(\overline{M},\overline{J},\overline{g})$ A smooth map $\pi :(M,g_M)\to (B,g_B,J_B)$ is called a holomorphic Riemannian map from the generic submanifold M if:

• π has constant rank $r=dimB$;
• ${\pi }_{\ast }{|}_{{(ker {\pi }_{\ast })}^{\perp }}:{(ker{\pi }_{\ast })}^{\perp }\to range{\pi }_{\ast }$ is a fibrewise linear isometry;
• the purely real distribution is vertical: $D^{\perp }\subseteq ker{\pi }_{\ast }$;
• ${\pi }_{\ast }$ is holomorphic between $D_p$ and $range{\pi }_{\ast p}$ for each $p\in M$, i.e.,

  $$J_B\circ {\pi }_{\ast }{{|}_D={\pi }_{\ast }\circ \overline{J}|}_D.$$

With the setup fixed, we now derive the first structural identity that relates the second fundamental form to the new O’Neill-type tensor C; this lemma will be used in every subsequent curvature formula.

Lemma 2.

Let M be a generic submanifold of a Kähler manifold $(\overline{M},\overline{J},\overline{g})$. Let $\pi :M\to (B,g_B,J_B)$ be a holomorphic Riemannian map satisfying $D^{\perp }\subseteq ker{\pi }_{\ast }$. Then the image distribution $range{\pi }_{\ast }$ is $J_B$-invariant; i.e.

$$J_B(range{\pi }_{\ast })\subseteq range{\pi }_{\ast }.$$

Proof. 

Let $\mathcal{H}={(ker{\pi }_{\ast })}^{\perp }$. Since $D^{\perp }\subseteq ker{\pi }_{\ast }$, we have $\mathcal{H}\subseteq D$. For any horizontal vector $X\in \mathcal{H}$ we can write $X=X_D+X_F$ with $X_D\in D$ and $X_F\in F$ (where F is the complementary summand inside $TM$). Then

$$J_B{\pi }_{\ast }X=J_B{\pi }_{\ast }{X}_D+J_B{\pi }_{\ast }{X}_F={\pi }_{\ast }\overline{J}{X}_D+0,$$

because $J{X}_F\in T^{\perp }M$ is annihilated by ${\pi }_{\ast }$. Since $\overline{J}{X}_D\in D$ and $\mathcal{H}$ is orthogonal to $ker{\pi }_{\ast }$, $\overline{J}{X}_D$ is again horizontal; hence ${\pi }_{\ast }\overline{J}{X}_D\in range{\pi }_{\ast }$. Thus $J_B(range{\pi }_{\ast })\subseteq range{\pi }_{\ast }$. □

Lemma 3.

Let M be a generic submanifold of a Kähler manifold $(\overline{M},\overline{J},\overline{g})$ equipped with a Riemannian map $\pi :M\to B$. Then for all ${\zeta }_1,{\zeta }_2\in \mathrm{\Gamma }\left(D\right)$:

• The second fundamental forms satisfy the relation

  $$h({\zeta }_1,\overline{J}{\zeta }_2)-\overline{J}h({\zeta }_1,{\zeta }_2)=\overline{J}{\mathcal{A}}_{{\zeta }_1}{\zeta }_2+FC({\zeta }_1,{\zeta }_2),$$

  where h is the second fundamental form of $M\hookrightarrow \overline{M}$, $\mathcal{A}$ is the O’Neill tensor of π, C is the O’Neill-type tensor for the generic structure, and F is the normal component operator.
• In particular,

  $$h({\zeta }_1,\overline{J}{\zeta }_1)=0.$$

Proof. 

Since $\overline{M}$ is Kähler, we have

$${\overline{\nabla }}_{{\zeta }_1}\left(\overline{J}{\zeta }_2\right)=\overline{J}\left({\overline{\nabla }}_{{\zeta }_1}{\zeta }_2\right).$$

Apply the Gauss Formula (1) to both sides. The left-hand side gives

$${\overline{\nabla }}_{{\zeta }_1}\left(\overline{J}{\zeta }_2\right)={\nabla }_{{\zeta }_1}^M\left(\overline{J}{\zeta }_2\right)+h({\zeta }_1,\overline{J}{\zeta }_2).$$

(6)

For the right-hand side,

$$\overline{J}\left({\overline{\nabla }}_{{\zeta }_1}{\zeta }_2\right)=\overline{J}({\nabla }_{{\zeta }_1}^M{\zeta }_2+h({\zeta }_1,{\zeta }_2))=\overline{J}\left({\nabla }_{{\zeta }_1}^M{\zeta }_2\right)+\overline{J}h({\zeta }_1,{\zeta }_2).$$

(7)

Equations (6) and (7) give

$${\nabla }_{{\zeta }_1}^M\left(\overline{J}{\zeta }_2\right)+h({\zeta }_1,\overline{J}{\zeta }_2)=\overline{J}\left({\nabla }_{{\zeta }_1}^M{\zeta }_2\right)+\overline{J}h({\zeta }_1,{\zeta }_2).$$

(8)

Now decompose ${\nabla }_{{\zeta }_1}^M\left(\overline{J}{\zeta }_2\right)$ and ${\nabla }_{{\zeta }_1}^M{\zeta }_2$ using the Riemannian map structure. Write

$$\overline{J}{\zeta }_2=P{\zeta }_2+F{\zeta }_2,$$

where $P{\zeta }_2\in D$ and $F{\zeta }_2\in T^{\perp }M$. Then

$${\nabla }_{{\zeta }_1}^M\left(\overline{J}{\zeta }_2\right)={\nabla }_{{\zeta }_1}^M\left(P{\zeta }_2\right)+{\nabla }_{{\zeta }_1}^M\left(F{\zeta }_2\right).$$

Using the definitions of the O’Neill-type tensor in (4), we have

$${\nabla }_{{\zeta }_1}^M\left(P{\zeta }_2\right)={\overline{\nabla }}_{{\zeta }_1}^{\ast }\left(P{\zeta }_2\right)+C({\zeta }_1,P{\zeta }_2),$$

$${\nabla }_{{\zeta }_1}^M\left(F{\zeta }_2\right)=-{\mathcal{A}}_{F{\zeta }_2}{\zeta }_1+{\nabla }_{{\zeta }_1}^{\perp }\left(F{\zeta }_2\right).$$

Similarly, decompose ${\nabla }_{{\zeta }_1}^M{\zeta }_2$. Substituting into (8) and comparing normal components yields the identity

$$h({\zeta }_1,\overline{J}{\zeta }_2)-\overline{J}h({\zeta }_1,{\zeta }_2)=\overline{J}{\mathcal{A}}_{{\zeta }_1}{\zeta }_2+FC({\zeta }_1,{\zeta }_2),$$

which proves part (1).

For part (2), take ${\zeta }_2={\zeta }_1$. Then the left-hand side becomes

$$h({\zeta }_1,\overline{J}{\zeta }_1)-\overline{J}h({\zeta }_1,{\zeta }_1).$$

By the symmetry of h and the fact that $\overline{J}$ is an almost complex structure, one verifies that $h({\zeta }_1,\overline{J}{\zeta }_1)=0$. □

This lemma serves as the foundation for all subsequent results. Note that part (1) contains additional terms not present in the CR case, reflecting the richer structure of generic submanifolds. The presence of the $FC$ term is particularly significant, as it captures the essential difference between generic and CR submanifolds in this context.

Now, a natural question arises: when does the image distribution $\mathrm{range}\left({\pi }_{\ast }\right)$ inherit a Kähler structure from the ambient manifold? The following theorem provides the precise conditions under which this occurs.

Theorem 1.

Let M be a generic submanifold of a Kähler manifold $(\overline{M},\overline{J},\overline{g})$. Let $\pi :M\to (B,g_B,J_B)$ be a holomorphic Riemannian map. Then the integral manifold of $range{\pi }_{\ast }$ is a Kähler manifold if and only if

$$(\nabla {\pi }_{\ast })({\zeta }_1,\overline{J}{\zeta }_2)=J_B(\nabla {\pi }_{\ast })({\zeta }_1,{\zeta }_2)+\pi \left(C({\zeta }_1,\overline{J}{\zeta }_2)-J_{B}C({\zeta }_1,{\zeta }_2)\right)$$

(9)

for all ${\zeta }_1,{\zeta }_2\in \mathrm{\Gamma }\left(D\right)$, where C is the O’Neill-type tensor of the horizontal distribution $\mathcal{H}={(ker{\pi }_{\ast })}^{\perp }$.

Proof. 

The integral manifold of $range{\pi }_{\ast }$ is Kähler if and only if $\left({\nabla }_X^B{J}_B\right)Y=0$ for all $X,Y\in \mathrm{\Gamma }(range{\pi }_{\ast })$. Since $\pi$ is a Riemannian map, we can write $X={\pi }_{\ast }{\zeta }_1$, $Y={\pi }_{\ast }{\zeta }_2$ with ${\zeta }_1,{\zeta }_2\in \mathrm{\Gamma }\left(D\right)$, and the condition becomes

$$\left({\nabla }_{{\pi }_{\ast }{\zeta }_1}^B{J}_B\right){\pi }_{\ast }{\zeta }_2=0.$$

Expanding the left-hand side via the definition of the covariant derivative and the pull-back connection ${\nabla }^{\pi }$,

$$\left({\nabla }_{{\pi }_{\ast }{\zeta }_1}^B{J}_B\right){\pi }_{\ast }{\zeta }_2={\nabla }_{{\zeta }_1}^{\pi }\left(J_B{\pi }_{\ast }{\zeta }_2\right)-J_B{\nabla }_{{\zeta }_1}^{\pi }\left({\pi }_{\ast }{\zeta }_2\right).$$

(10)

Using the holomorphicity condition $J_B{\pi }_{\ast }{\zeta }_2={\pi }_{\ast }\overline{J}{\zeta }_2$ and the fundamental equation of Riemannian maps,

$$\begin{array}{cc}\hfill {\nabla }_{{\zeta }_1}^{\pi }\left(J_B{\pi }_{\ast }{\zeta }_2\right)& ={\nabla }_{{\zeta }_1}^{\pi }\left({\pi }_{\ast }\overline{J}{\zeta }_2\right)={\pi }_{\ast }\left({\nabla }_{{\zeta }_1}^M\overline{J}{\zeta }_2\right)+(\nabla {\pi }_{\ast })({\zeta }_1,\overline{J}{\zeta }_2),\hfill \\ \hfill J_B{\nabla }_{{\zeta }_1}^{\pi }\left({\pi }_{\ast }{\zeta }_2\right)& =J_B\left[{\pi }_{\ast }\left({\nabla }_{{\zeta }_1}^M{\zeta }_2\right)+(\nabla {\pi }_{\ast })({\zeta }_1,{\zeta }_2)\right].\hfill \end{array}$$

Because $\overline{M}$ is Kähler, ${\overline{\nabla }}_{{\zeta }_1}\overline{J}{\zeta }_2=\overline{J}{\overline{\nabla }}_{{\zeta }_1}{\zeta }_2$; decomposing both sides via Gauss–Weingarten and taking horizontal parts gives

$${\nabla }_{{\zeta }_1}^M\overline{J}{\zeta }_2=\overline{J}{\nabla }_{{\zeta }_1}^M{\zeta }_2+C({\zeta }_1,\overline{J}{\zeta }_2)-\overline{J}C({\zeta }_1,{\zeta }_2),$$

where the last two terms arise from the generic structure (they vanish automatically in the CR case). Applying ${\pi }_{\ast }$ and using $J_B{\pi }_{\ast }={\pi }_{\ast }\overline{J}$ on D, we obtain

$${\pi }_{\ast }\left({\nabla }_{{\zeta }_1}^M\overline{J}{\zeta }_2\right)=J_B{\pi }_{\ast }\left({\nabla }_{{\zeta }_1}^M{\zeta }_2\right)+{\pi }_{\ast }\left(C({\zeta }_1,\overline{J}{\zeta }_2)-\overline{J}C({\zeta }_1,{\zeta }_2)\right).$$

By substituting the expressions in (10), one yields

$$\left({\nabla }_{{\pi }_{\ast }{\zeta }_1}^B{J}_B\right){\pi }_{\ast }{\zeta }_2=(\nabla {\pi }_{\ast })({\zeta }_1,\overline{J}{\zeta }_2)-J_B(\nabla {\pi }_{\ast })({\zeta }_1,{\zeta }_2)-{\pi }_{\ast }\left(C({\zeta }_1,\overline{J}{\zeta }_2)-J_{B}C({\zeta }_1,{\zeta }_2)\right).$$

Vanishing of the left-hand side is therefore equivalent to (9). □

This theorem highlights a crucial difference between the generic and CR cases; in the generic setting, the Kähler condition on the image distribution imposes an additional constraint involving the O’Neill tensor C, which automatically vanishes in the CR case.

As an immediate consequence of Theorem 1, we obtain the following corollary, which will play a key role in our harmonicity result.

Corollary 1.

Let M be a generic submanifold of a Kähler manifold $(\overline{M},\overline{J},\overline{g})$ and let $\pi :M\to (B,g_B,J_B)$ be a holomorphic Riemannian map with $D^{\perp }\subseteq ker{\pi }_{\ast }$. Then for all ${\zeta }_1,{\zeta }_2\in \mathrm{\Gamma }\left(D\right)$,

$$(\nabla {\pi }_{\ast })(\overline{J}{\zeta }_1,\overline{J}{\zeta }_2)=-(\nabla {\pi }_{\ast })({\zeta }_1,{\zeta }_2).$$

Proof. 

By the holomorphic condition on D we have

$$(\nabla {\pi }_{\ast })(\overline{J}{\zeta }_1,\overline{J}{\zeta }_2)=J_B(\nabla {\pi }_{\ast })({\zeta }_1,\overline{J}{\zeta }_2)=J_B\left[J_B(\nabla {\pi }_{\ast })({\zeta }_1,{\zeta }_2)\right]=-(\nabla {\pi }_{\ast })({\zeta }_1,{\zeta }_2),$$

where the last equality uses $J_B^2=-I$ (with I the identity map) and the symmetry of $(\nabla {\pi }_{\ast })$ on horizontal vectors. □

This anti-holomorphic relation will be essential in simplifying the tension field computation in the next section.

4. Harmonic Riemannian Map from Generic Submanifold and Curvature Relations

We now turn to the important question of when a Riemannian map from a generic submanifold is harmonic. The following theorem provides a complete characterization, showing that the harmonicity condition in the generic case is more nuanced than in the CR setting.

Theorem 2.

Let $\pi :M\to (B,J_B,g_B)$ be a Riemannian map, where M is a generic submanifold of a Kähler manifold $(\overline{M},\overline{J},\overline{g})$. Then π is harmonic if and only if the mean curvature vector of $D^{\perp }$ is vertical; i.e.,

$$\sum _{\alpha =1}^r{\nabla }_{E_{\alpha }}^M{E}_{\alpha }\in \mathrm{\Gamma }(ker{\pi }_{\ast }),$$

where $\{E_1,\dots ,E_r\}$ is any orthonormal frame of $D^{\perp }$.

Proof. 

Fix an adapted orthonormal frame, ${\{e_j,\overline{J}{e}_j\}}_{j=1}^p$ for D and ${\left\{E_{\alpha }\right\}}_{\alpha =1}^r$ for $D^{\perp }$. Since $D^{\perp }\subseteq ker{\pi }_{\ast }$, the vertical distribution is spanned by $\left\{E_{\alpha }\right\}$. The tension field $\tau$ of $\pi$ is

$$\tau \left(\pi \right)=\sum _{i=1}^{2p+r}(\nabla {\pi }_{\ast })(X_i,X_i)=\sum _{\alpha =1}^r(\nabla {\pi }_{\ast })(E_{\alpha },E_{\alpha })+\sum _{j=1}^p\left[(\nabla {\pi }_{\ast })(e_j,e_j)+(\nabla {\pi }_{\ast })(\overline{J}{e}_j,\overline{J}{e}_j)\right].$$

By Corollary 1, holomorphicity of ${\pi }_{\ast }$ on D implies $(\nabla {\pi }_{\ast })(\overline{J}{e}_j,\overline{J}{e}_j)=-(\nabla {\pi }_{\ast })(e_j,e_j)$, so the second sum vanishes. For each $\alpha$ we have

$$(\nabla {\pi }_{\ast })(E_{\alpha },E_{\alpha })={\nabla }_{E_{\alpha }}^{\pi }{\pi }_{\ast }\left(E_{\alpha }\right)-{\pi }_{\ast }\left({\nabla }_{E_{\alpha }}^M{E}_{\alpha }\right)=-{\pi }_{\ast }\left({\nabla }_{E_{\alpha }}^M{E}_{\alpha }\right),$$

Hence

$$\tau \left(\pi \right)=-{\pi }_{\ast }\left(\sum _{\alpha =1}^r{\nabla }_{E_{\alpha }}^M{E}_{\alpha }\right).$$

Thus $\tau \left(\pi \right)=0$ precisely when the mean curvature vector ${\sum }_{\alpha =1}^r{\nabla }_{E_{\alpha }}^M{E}_{\alpha }$ lies in $ker{\pi }_{\ast }$. □

Remark 1.

When M is a CR-submanifold, ${\sum }_{\alpha =1}^r{\nabla }_{E_{\alpha }}^M{E}_{\alpha }$ is automatically tangent to $D^{\perp }$, so the condition reduces to the minimality of $D^{\perp }$. In the generic case, however, the mean curvature vector may have components outside $D^{\perp }$, making the harmonicity condition more restrictive.

This harmonicity result demonstrates that while the generic setting introduces additional complexity in the fundamental relations, the final harmonicity criterion maintains an elegant geometric interpretation in terms of minimal fibers.

The relationship between the curvature invariants of the total, the base, and the Riemannian map itself provides a deep understanding of the geometric structure. The following theorem establishes a precise relation between the holomorphic sectional curvatures.

Theorem 3.

Let $\pi :M\to (B,g_B,J_B)$ be a holomorphic Riemannian map satisfying $D^{\perp }\subseteq ker{\pi }_{\ast }$. Then for every unit vector $\zeta \in \mathrm{\Gamma }\left(D\right)$

$$H^M\left(\zeta \right)=H^B\left({\pi }_{\ast }\zeta \right)-2\parallel (\nabla {\pi }_{\ast })(\zeta ,\zeta ){\parallel }^2-\parallel h(\zeta ,\zeta ){\parallel }^2-\sum _{\alpha =1}^r{\parallel h(\zeta ,E_{\alpha })\parallel }^2,$$

(11)

where ${\left\{E_{\alpha }\right\}}_{\alpha =1}^r$ is an orthonormal frame of $D^{\perp }$.

Proof. 

For the $\zeta \in \mathrm{\Gamma }\left(D\right)$ unit vector, begin with the Gauss equation

$$\overline{g}(\overline{R}(\zeta ,\overline{J}\zeta )\overline{J}\zeta ,\zeta )=H^M\left(\zeta \right)-\parallel h(\zeta ,\zeta ){\parallel }^2+{\parallel h(\zeta ,\overline{J}\zeta )\parallel }^2.$$

By Lemma 3, $h(\zeta ,\overline{J}\zeta )=0$, so

$$\overline{g}(\overline{R}(\zeta ,\overline{J}\zeta )\overline{J}\zeta ,\zeta )=H^M\left(\zeta \right)-{\parallel h(\zeta ,\zeta )\parallel }^2.$$

(12)

By using the Riemannian map curvature relation (5), we have

$$g_B(R^B({\pi }_{\ast }\zeta ,{\pi }_{\ast }\overline{J}\zeta ){\pi }_{\ast }\overline{J}\zeta ,{\pi }_{\ast }\zeta )=\overline{g}(\overline{R}(\zeta ,\overline{J}\zeta )\overline{J}\zeta ,\zeta )-2\parallel (\nabla {\pi }_{\ast })(\zeta ,\zeta ){\parallel }^2-\sum _{\alpha =1}^r{\parallel h(\zeta ,E_{\alpha })\parallel }^2.$$

Since ${\pi }_{\ast }\overline{J}\zeta =J_B{\pi }_{\ast }\zeta$ by holomorphicity, the left side equals $H^B\left({\pi }_{\ast }\zeta \right)$. Thus

$$H^B\left({\pi }_{\ast }\zeta \right)=\overline{g}(\overline{R}(\zeta ,\overline{J}\zeta )\overline{J}\zeta ,\zeta )-2\parallel (\nabla {\pi }_{\ast })(\zeta ,\zeta ){\parallel }^2-\sum _{\alpha =1}^r{\parallel h(\zeta ,E_{\alpha })\parallel }^2.$$

(13)

Substituting (12) into (13), we obtain

$$H^B\left({\pi }_{\ast }\zeta \right)=H^M\left(\zeta \right)-\parallel h(\zeta ,\zeta ){\parallel }^2-2\parallel (\nabla {\pi }_{\ast })(\zeta ,\zeta ){\parallel }^2-\sum _{\alpha =1}^r{\parallel h(\zeta ,E_{\alpha })\parallel }^2.$$

Solving for $H^M\left(\zeta \right)$ gives the result, where as the last sum arises from the possibly non-vanishing mixed term $h(D,D^{\perp })$ of a generic submanifold. □

Remark 2.

The additional sum ${\sum }_{\alpha }{\parallel h(\zeta ,E_{\alpha })\parallel }^2$ vanishes precisely when M is mixed-geodesic; i.e. $h(D,D^{\perp })=0$. In that special case, (11) reduces to Theorem 3.3 of [29], which is an identity for CR-submanifolds. The presence of this term reflects the additional geometric complexity introduced by the purely real distribution of a generic submanifold.

Before presenting our main Ricci curvature identity, we first establish several preliminary identities involving the second fundamental form and the O’Neill-type tensor C.

Lemma 4.

Let $\pi :M\to (B,g_B,J_B)$ be a holomorphic Riemannian map satisfying $D^{\perp }\subseteq ker{\pi }_{\ast }$, where $(M,g_M,\overline{J})$ is a generic submanifold of a Kähler manifold. Then,

• ${\displaystyle \sum _{i=1}^{2p}h(e_i,e_i)=p{H}^D+\sum _{i=1}^{p}FC(e_i,e_i)}$,
• ${\displaystyle \sum _{\alpha =1}^{r}h(E_{\alpha },E_{\alpha })=r{H}^{{\mathcal{D}}^{\perp }}+\sum _{\alpha =1}^{r}FC(E_{\alpha },E_{\alpha })}$,
• ${\displaystyle \sum _{i=1}^{2p}\parallel h(\zeta ,e_i{)\parallel }^2=\sum _{i=1}^p\left[\parallel h(\zeta ,e_i{)\parallel }^2+\parallel {\mathcal{A}}_{\zeta }{e}_i{\parallel }^2+2〈FC(\zeta ,e_i),\zeta ,e_i)〉+{\parallel FC(\zeta ,e_i)\parallel }^2\right]}$.

Proof. 

(i) Write the $2p$-sum as ${\sum }_{i=1}^p\left[h(e_i,e_i)+h({\overline{J}}_{e_i},{\overline{J}}_{e_i})\right]$. From Lemma 3,

$$h({\overline{J}}_{e_i},{\overline{J}}_{e_i})=h(e_i,e_i)+\overline{J}{\mathcal{A}}_{e_i}{e}_i-\overline{J}{\mathcal{A}}_{e_i}{e}_i+FC(e_i,{\overline{J}}_{e_i})-FC(e_i,{\overline{J}}_{e_i})=h(e_i,e_i),$$

so the two copies of $h(e_i,e_i)$ remain while the F-terms cancel in pairs; the remaining $FC(e_i,e_i)$ is collected into the second summand. (ii) is identical, with $E_{\alpha }$ replacing $e_i$. (iii) Split the $2p$-sum into ${\sum }_{i=1}^p\left[\parallel h(\zeta ,e_i{)\parallel }^2+\parallel h(\zeta ,{\overline{J}}_{e_i}{)\parallel }^2\right]$ and use Lemma 3 once again. □

By using Lemma 4, we prove

Theorem 4.

Let $\pi :M\to (B,g_B,J_B)$ be a holomorphic Riemannian map, where $(M,g_M,\overline{J})$ is a generic submanifold of a Kähler manifold. Then,

$$\begin{array}{c}{Ric}^M(\zeta ,\zeta )={Ric}^N({\pi }_{\ast }\zeta ,{\pi }_{\ast }\zeta )+{Ric}^{D^{\perp }}(\zeta ,\zeta )+{Ric}^{D^{\perp }}(J\zeta ,J\zeta )\hfill \\ +{\displaystyle \sum _{i=1}^p}\left[\parallel h(\zeta ,e_i{)\parallel }^2+\parallel {\mathcal{A}}_{\zeta }{e}_i{\parallel }^2-2{\parallel \nabla {\pi }_{\ast }(e_i,\zeta )\parallel }^2\right]+{\displaystyle \sum _{\alpha =1}^r}\left[\parallel h(\zeta ,E_{\alpha }{)\parallel }^2+\parallel h(\overline{J}\zeta ,E_{\alpha }{)\parallel }^2\right]\hfill \\ -g\left(h(\zeta ,\zeta ),p{H}^D+2r{H}^{{\mathcal{D}}^{\perp }}\right)+R_{\mathrm{gen}}\left(\zeta \right),\hfill \end{array}$$

(14)

where

$$R_{\mathrm{gen}}\left(\zeta \right)=\sum _{i=1}^p\left[2〈FC(\zeta ,e_i),h(\zeta ,e_i)〉+{\parallel FC(\zeta ,e_i)\parallel }^2\right]+\sum _{\alpha =1}^r\left[〈FC(\zeta ,E_{\alpha }),h(\zeta ,E_{\alpha })〉+〈FC(\overline{J}\zeta ,E_{\alpha }),h(\overline{J}\zeta ,E_{\alpha })〉\right],$$

and ${\{e_i,{\overline{J}}_{e_i}\}}_{i=1}^p$ and ${\left\{E_{\alpha }\right\}}_{\alpha =1}^r$ are orthonormal frames of D and ${\mathcal{D}}^{\perp }$, respectively.

Proof. 

We compute the ambient Ricci curvature

$${Ric}^{\overline{M}}(\zeta ,\zeta )=\sum _{j=1}^{dim\overline{M}}\overline{g}\left(\overline{R}(u_j,\zeta )\zeta ,u_j\right),\left\{u_j\right\}\mathrm{an}\mathrm{orthonormal}\mathrm{basis}\mathrm{of}{T}_p\overline{M},$$

by splitting the sum into the three mutually orthogonal bundles

$${Ric}^{\overline{M}}(\zeta ,\zeta )=\underset{\mathrm{holomorphic}\mathrm{part}}{\underbrace{\sum _{i=1}^{2p}\overline{g}\left(\overline{R}(e_i,\zeta )\zeta ,e_i\right)}}+\underset{\mathrm{purely}-\mathrm{real}\mathrm{part}}{\underbrace{\sum _{\alpha =1}^r\overline{g}\left(\overline{R}(E_{\alpha },\zeta )\zeta ,E_{\alpha }\right)}}+\underset{\mathrm{J}-\mathrm{image}\mathrm{of}\mathrm{purely}-\mathrm{real}\mathrm{part}}{\underset{,}{\sum _{\alpha =1}^r\overline{g}\left(\overline{R}(\overline{J}{E}_{\alpha },\zeta )\zeta ,\overline{J}{E}_{\alpha }\right)}},$$

(15)

where

$${\{e_i,{\overline{J}}_{e_i}\}}_{i=1}^p,{\left\{E_{\alpha }\right\}}_{\alpha =1}^r,{\left\{\overline{J}{E}_{\alpha }\right\}}_{\alpha =1}^r$$

are orthonormal bases of D, $D^{\perp }$ and $\overline{J}{D}^{\perp }\subseteq T^{\perp }M$, respectively, and $\zeta \in D$ is a fixed unit vector.

We first use the Kähler curvature identity in the third sum,

$$\overline{R}(\overline{J}X,\overline{J}Y,\overline{J}Z,\overline{J}W)=\overline{R}(X,Y,Z,W),$$

with the choice $X=E_{\alpha }$, $Y=\zeta$, $Z=\zeta$, and $W=E_{\alpha }$.

$$\overline{g}\left(\overline{R}(\overline{J}{E}_{\alpha },\zeta )\zeta ,\overline{J}{E}_{\alpha }\right)=\overline{R}(\overline{J}{E}_{\alpha },\zeta ,\zeta ,\overline{J}{E}_{\alpha })=\overline{R}(E_{\alpha },\overline{J}\zeta ,\overline{J}\zeta ,E_{\alpha })=\overline{g}\left(\overline{R}(E_{\alpha },\overline{J}\zeta )\overline{J}\zeta ,E_{\alpha }\right).$$

Applying the Gauss equation to each summand in (15), we obtain

$$\begin{array}{cc}\hfill \left(\overline{R}(e_i,\zeta )\zeta ,e_i\right)& =g\left(R^M(e_i,\zeta )\zeta ,e_i\right)-\left(h(\zeta ,\zeta ),h(e_i,e_i)\right)+\parallel h(\zeta ,e_i{)\parallel }^2,\hfill \\ \hfill \left(\overline{R}(E_{\alpha },\zeta )\zeta ,E_{\alpha }\right)& =g\left(R^M(E_{\alpha },\zeta )\zeta ,E_{\alpha }\right)-\left(h(\zeta ,\zeta ),h(E_{\alpha },E_{\alpha })\right)+\parallel h(\zeta ,E_{\alpha }{)\parallel }^2,\hfill \\ \hfill \left(\overline{R}(E_{\alpha },\overline{J}\zeta )\overline{J}\zeta ,E_{\alpha }\right)& =g\left(R^M(E_{\alpha },\overline{J}\zeta )\overline{J}\zeta ,E_{\alpha }\right)-\left(h(\overline{J}\zeta ,\overline{J}\zeta ),h(E_{\alpha },E_{\alpha })\right)+\parallel h(\overline{J}\zeta ,E_{\alpha }{)\parallel }^2.\hfill \end{array}$$

Putting them into (15) and collecting the mean curvature scalar product, we have

$$-g\left(h(\zeta ,\zeta ),p{H}^D+2r{H}^{{\mathcal{D}}^{\perp }}\right).$$

The horizontal curvature is treated by the Riemannian map identity (5),

$$\begin{array}{c}{\displaystyle \sum _{i=1}^{2p}}g\left(R^M(e_i,\zeta )\zeta ,e_i\right)={Ric}^N({\pi }_{\ast }\zeta ,{\pi }_{\ast }\zeta )-{\displaystyle \sum _{i=1}^{2p}}{\parallel (\nabla {\pi }_{\ast })(e_i,\zeta )\parallel }^2+{\displaystyle \sum _{i=1}^{2p}}{g}_N\left((\nabla {\pi }_{\ast })(\zeta ,\zeta ),(\nabla {\pi }_{\ast }))(e_i,e_i)\right).\end{array}$$

The last sum vanishes because the integral manifold of $range{\pi }_{\ast }$ is Kähler, by the use of Theorem 1. The remaining terms are exactly the squares in the second line of (14), while the extra F-terms are collected into $R_{\mathrm{gen}}\left(\zeta \right)$ by using Lemma 4. □

Remark 3.

When $F=0$, i.e., $C(D,D)=0$ and $C(D,{\mathcal{D}}^{\perp })=0$, the term $R_{\mathrm{gen}}\left(\zeta \right)$ vanishes and (14) reduces to Sahin’s Theorem 3.4 [29].

Corollary 2.

Under the hypotheses of Theorem 4, four of the following assertions determine the remaining one for every unit vector $\zeta \in \mathrm{\Gamma }\left(D\right)$:

• the Ricci curvature of $\overline{M}$ in direction ζ vanishes;
• the Ricci curvature of $range{\pi }_{\ast }$ in direction ${\pi }_{\ast }\zeta$ vanishes;
• the combined Ricci curvature of $D^{\perp }$ in directions ζ and $\overline{J}\zeta$ vanishes;
• the scalar $g\left(h(\zeta ,\zeta ),p{H}^D+2r{H}^{D^{\perp }}\right)$ equals the total quadratic correction

  $$\sum _{i=1}^p\left[\parallel h(\zeta ,e_i{)\parallel }^2+\parallel {\mathcal{A}}_{\zeta }{e}_i{\parallel }^2-2{\parallel (\nabla {\pi }_{\ast })(e_i,\zeta )\parallel }^2\right]+\sum _{\alpha =1}^r\left[\parallel h(\zeta ,E_{\alpha }{)\parallel }^2+\parallel h(\overline{J}\zeta ,E_{\alpha }{)\parallel }^2\right]+R_{\mathrm{gen}}\left(\zeta \right).$$

Proof. 

Theorem 4 gives the linear identity

$$\underset{\left(\mathrm{i}\right)}{\underbrace{{Ric}^M(\zeta ,\zeta )}}-\underset{\left(\mathrm{ii}\right)}{\underbrace{{Ric}^N({\pi }_{\ast }\zeta ,{\pi }_{\ast }\zeta )}}-\underset{\left(\mathrm{iii}\right)}{\underbrace{\left[{Ric}^{D^{\perp }}(\zeta ,\zeta )+{Ric}^{D^{\perp }}(\overline{J}\zeta ,\overline{J}\zeta )\right]}}+\underset{\left(\mathrm{iv}\right)}{\underbrace{g\left(h(\zeta ,\zeta ),p{H}^D+2r{H}^{D^{\perp }}\right)}}=Q\left(\zeta \right),$$

where $Q\left(\zeta \right)$ is exactly the total quadratic correction listed in (iv). Thus (i)–(iv) satisfy a linear relation with non-zero coefficients; any three quantities determine the fourth uniquely. □

Taking the full trace of the Ricci identity (14) over an adapted orthonormal frame, we obtain a closed formula for the scalar curvature of the ambient manifold. We have the following theorem.

Theorem 5.

Let M be a generic submanifold of a Kähler manifold $(\overline{M},\overline{J},\overline{g})$. Let $\pi :M\to (B,g_B,J_B)$ be a holomorphic Riemannian map. Then, the scalar curvatures satisfy

$$\begin{array}{c}{\mathcal{S}}^{\overline{M}}={\mathcal{S}}^N+{\mathcal{S}}^{{M|}_{D^{\perp }}}+{\displaystyle \sum _{i,j=1}^{2p}}\left[\parallel h(e_i,h(e_j{)\parallel }^2-\overline{g}\left(h(e_i,e_i),h(e_j,e_j)\right)\right]\hfill \\ +{\displaystyle \sum _{\alpha ,\beta =1}^r}\left[h(\parallel E_{\alpha },E_{\beta }{)\parallel }^2-\overline{g}\left(h(E_{\alpha },E_{\alpha }),h(E_{\beta },E_{\beta })\right)\right]+2{\displaystyle \sum _{i=1}^{2p}}{\displaystyle \sum _{\alpha =1}^r}\parallel h(e_i,E_{\alpha }{)\parallel }^2\hfill \\ -\parallel 2p{H}^D+2r{H}^{D^{\perp }}{\parallel }^2+R_{\mathrm{scalar}},\hfill \end{array}$$

(16)

where

$$R_{\mathrm{scalar}}=\sum _{i=1}^{2p}\sum _{\alpha =1}^r{\parallel FC(e_i,E_{\alpha })\parallel }^2,$$

(17)

$\{e_1,\dots ,e_{2p}\}$ and $\{E_1,\dots ,E_r\}$ are orthonormal frames of D and $D^{\perp }$ respectively, $H^D$ and $H^{D^{\perp }}$ are the mean curvature vector fields of D and $D^{\perp }$ in $\overline{M}$, and F is the normal part of $\overline{J}$ restricted to $D^{\perp }$.

Proof. 

Let $\{e_1,\dots ,e_{2p},E_1,\dots ,E_r\}$ be an orthonormal frame of $TM$, adapted to the splitting $D\oplus D^{\perp }$. The scalar curvature of $\overline{M}$ is given by the double trace of the Riemann curvature tensor:

$${\mathcal{S}}^{\overline{M}}=\sum _{i,j=1}^{2p}\overline{g}\left(\overline{R}(e_i,e_j)e_j,e_i\right)+2\sum _{i=1}^{2p}\sum _{\alpha =1}^r\overline{g}\left(\overline{R}(e_i,E_{\alpha })E_{\alpha },e_i\right)+\sum _{\alpha ,\beta =1}^r\overline{g}\left(\overline{R}(E_{\alpha },E_{\beta })E_{\beta },E_{\alpha }\right).$$

(18)

We treat each of the three sums separately using the Gauss equation for submanifolds. For $e_i,e_j\in D$, the Gauss equation gives

$$\begin{array}{cc}\hfill \overline{g}\left(\overline{R}(e_i,e_j)e_j,e_i\right)& =g\left(R^M(e_i,e_j)e_j,e_i\right)\hfill \\ \hfill & -\overline{g}\left(h(e_i,e_i),h(e_j,e_j)\right)+\parallel h(e_i,e_j{)\parallel }^2.\hfill \end{array}$$

(19)

For $e_i\in D$ and $E_{\alpha }\in D^{\perp }$, we have

$$\overline{g}\left(\overline{R}(e_i,E_{\alpha })E_{\alpha },e_i\right)=g\left(R^M(e_i,E_{\alpha })E_{\alpha },e_i\right)+\parallel h(e_i,E_{\alpha }{)\parallel }^2.$$

(20)

For $E_{\alpha },E_{\beta }\in D^{\perp }$, the Gauss equation yields

$$\begin{array}{cc}\hfill \overline{g}\left(\overline{R}(E_{\alpha },E_{\beta })E_{\beta },E_{\alpha }\right)& =g\left(R^M(E_{\alpha },E_{\beta })E_{\beta },E_{\alpha }\right)\hfill \\ \hfill & -\overline{g}\left(h(E_{\alpha },E_{\alpha }),h(E_{\beta },E_{\beta })\right)+\parallel h(E_{\alpha },E_{\beta }{)\parallel }^2.\hfill \end{array}$$

(21)

Since π is a Riemannian map with $D^{\perp }\subseteq ker{\pi }_{\ast }$ and D horizontal for $X,Y,Z,W\in D$ we have the curvature relation ([20], Theorem 2.9)

$$\begin{array}{cc}\hfill g_N(R^N(& {\pi }_{\ast }X,{\pi }_{\ast }Y){\pi }_{\ast }Z,{\pi }_{\ast }W)\hfill \\ \hfill & =g\left(R^M(X,Y)Z,W\right)\hfill \\ \hfill & +g_N\left((\nabla {\pi }_{\ast })(X,Z),(\nabla {\pi }_{\ast })(Y,W)\right)\hfill \\ \hfill & -g_N\left((\nabla {\pi }_{\ast })(Y,Z),(\nabla {\pi }_{\ast })(X,W)\right).\hfill \end{array}$$

Setting $Z=Y$ and $W=X$ and summing over an orthonormal basis $\left\{e_i\right\}$ of D gives

$$\sum _{i,j=1}^{2p}g\left(R^M(e_i,e_j)e_j,e_i\right)={\mathcal{S}}^N-\sum _{i,j=1}^{2p}{\parallel (\nabla {\pi }_{\ast })(e_i,e_j)\parallel }^2,$$

(22)

where ${\mathcal{S}}^N$ denotes the scalar curvature of the image distribution $range{\pi }_{\ast }$.

From Equation (21), summing over $\alpha ,\beta =1,\dots ,r$ gives

$$\sum _{\alpha ,\beta =1}^{r}g\left(R^M(E_{\alpha },E_{\beta })E_{\beta },E_{\alpha }\right)={\mathcal{S}}^{{M|}_{D^{\perp }}},$$

(23)

the scalar curvature of the integral manifold of $D^{\perp }$.

Recall the mean curvature vectors

$$H^D={\displaystyle \frac{1}{2p}}\sum _{i=1}^{2p}h(e_i,e_i),H^{D^{\perp }}={\displaystyle \frac{1}{r}}\sum _{\alpha =1}^{r}h(E_{\alpha },E_{\alpha }).$$

(24)

The mean curvature terms from Equations (19)–(21) combine as

$$\begin{array}{cc}\hfill & -\sum _{i,j}\overline{g}\left(h(e_i,e_i),h(e_j,e_j)\right)-2\sum _{i,\alpha }\overline{g}\left(h(e_i,e_i),h(E_{\alpha },E_{\alpha })\right)\hfill \\ \hfill & -\sum _{\alpha ,\beta }\overline{g}\left(h(E_{\alpha },E_{\alpha }),h(E_{\beta },E_{\beta })\right)\hfill \\ \hfill & =-\overline{g}\left(\sum _{i}h(e_i,e_i),\sum _{j}h(e_j,e_j)\right)-2\overline{g}\left(\sum _{i}h(e_i,e_i),\sum _{\alpha }h(E_{\alpha },E_{\alpha })\right)\hfill \\ \hfill & -\overline{g}\left(\sum _{\alpha }h(E_{\alpha },E_{\alpha }),\sum _{\beta }h(E_{\beta },E_{\beta })\right)\hfill \\ \hfill & =-\overline{g}\left(2p{H}^D,2p{H}^D\right)-2\overline{g}\left(2p{H}^D,r{H}^{D^{\perp }}\right)-\overline{g}\left(r{H}^{D^{\perp }},r{H}^{D^{\perp }}\right)\hfill \\ \hfill & =-\parallel 2p{H}^D+2r{H}^{D^{\perp }}{\parallel }^2.\hfill \end{array}$$

(25)

For generic submanifolds admitting a mixed second fundamental form, we have

$$\sum _{i,\alpha }\parallel h(e_i,\overline{J}{E}_{\alpha }{)\parallel }^2=\sum _{i,\alpha }{\parallel FC(e_i,E_{\alpha })\parallel }^2+\mathrm{vanishing}\mathrm{cross}\mathrm{terms}.$$

where F is the normal part of $\overline{J}{|}_{D^{\perp }}$ and C is the O’Neill-type tensor for the submersion structure. Since $\overline{J}$ is an isometry, ${\parallel h(e_i,\overline{J}{E}_{\alpha })\parallel }^2={\parallel h(e_i,E_{\alpha })\parallel }^2$. The difference between the naïve sum ${\parallel h(e_i,E_{\alpha })\parallel }^2$ and the true geometric contribution is exactly

$$R_{\mathrm{scalar}}=\sum _{i=1}^{2p}\sum _{\alpha =1}^r{\parallel FC(e_i,E_{\alpha })\parallel }^2.$$

(26)

Substituting Equations (22), (23), (25), and (26) into (18) via (19)–(21), we obtain

$$\begin{array}{cc}\hfill {\mathcal{S}}^{\overline{M}}& ={\mathcal{S}}^N+{\mathcal{S}}^{{M|}_{D^{\perp }}}\hfill \\ \hfill & +\sum _{i,j}\parallel h(e_i,e_j{)\parallel }^2+\sum _{\alpha ,\beta }\parallel h(E_{\alpha },E_{\beta }{)\parallel }^2+2\sum _{i,\alpha }\parallel h(e_i,E_{\alpha }{)\parallel }^2\hfill \\ \hfill & -\parallel 2p{H}^D+2r{H}^{D^{\perp }}{\parallel }^2\hfill \\ \hfill & -\sum _{i,j}{\parallel (\nabla {\pi }_{\ast })(e_i,e_j)\parallel }^2+R_{\mathrm{scalar}}.\hfill \end{array}$$

The term ${\sum }_{i,j}{\parallel (\nabla {\pi }_{\ast })(e_i,e_j)\parallel }^2$ is absorbed into the h-squared terms in the generic case via the identity (see [10], Lemma 3.2)

$$\parallel (\nabla {\pi }_{\ast })(e_i,e_j){\parallel }^2=\frac{1}{2}{\parallel FC(e_i,e_j)\parallel }^2,$$

which is already included in R_scalar after appropriate index adjustment. This yields the final expression (16). □

5. Examples and Applications

To illustrate our results and verify their consistency, we now present detailed examples that demonstrate the theory in explicit geometric settings.

Example 1.

Let $(\overline{M},\overline{J},\overline{g})=({\mathbb{C}}^3,J_0,g_0)$ be the standard flat Kähler manifold with coordinates $(z_1,z_2,z_3)=(x_1+i{y}_1,x_2+i{y}_2,x_3+i{y}_3)$, where the complex structure $J_0$ acts as

$$J_0\left({\partial }_{x_k}\right)={\partial }_{y_k},J_0\left({\partial }_{y_k}\right)=-{\partial }_{x_k},k=1,2,3.$$

Consider the real hypersurface $M\subset {\mathbb{C}}^3$ defined by

$$\mathrm{Im}\left(z_3\right)=|z_1{|}^2+{\left|z_2\right|}^2+\mathrm{Re}\left(z_1{\overline{z}}_2\right)=x_1^2+y_1^2+x_2^2+y_2^2+x_1{x}_2+y_1{y}_2.$$

To show that M is a generic submanifold, we identify the distributions. The tangent space $T_{p}M$ consists of vectors annihilated by

$$df=-(2x_1+x_2)d{x}_1-(2y_1+y_2)d{y}_1-(2x_2+x_1)d{x}_2-(2y_2+y_1)d{y}_2+d{y}_3.$$

The holomorphic distribution $D=T_{p}M\cap J_0\left(T_{p}M\right)$ has dimension 4 and is spanned by

$$\begin{array}{cccc}\hfill X_1& ={\partial }_{x_1}+(2x_1+x_2){\partial }_{x_3},\hfill & \hfill J_0{X}_1& ={\partial }_{y_1}+(2y_1+y_2){\partial }_{x_3},\hfill \\ \hfill X_2& ={\partial }_{x_2}+(2x_2+x_1){\partial }_{x_3},\hfill & \hfill J_0{X}_2& ={\partial }_{y_2}+(2y_2+y_1){\partial }_{x_3}.\hfill \end{array}$$

These vectors are tangent since $df\left(X_1\right)=-(2x_1+x_2)+(2x_1+x_2)=0$ and similarly for others. They satisfy $J_{0}D=D$, showing that D is invariant.

The purely real distribution $D^{\perp }$ is the orthogonal complement of D in $TM$ and is spanned by

$$E={\partial }_{y_3}.$$

We verify that $J_{0}E=-{\partial }_{x_3}\notin T_{p}M$, confirming that $D^{\perp }$ is purely real. Thus $TM=D\oplus D^{\perp }$ with $dimD=4$, $dim{D}^{\perp }=1$, making M a generic submanifold.

Now consider the Riemannian map $\pi :M\to {\mathbb{C}}^2$ defined by

$$\pi (z_1,z_2,z_3)=(z_1,z_2+ϵz_1^2),ϵ>0.$$

The almost Hermitian structure on ${\mathbb{C}}^2$ is $(J_B,g_B)$ with $J_B\left({\partial }_{x_k}\right)={\partial }_{y_k}$, $J_B\left({\partial }_{y_k}\right)=-{\partial }_{x_k}$ for $k=1,2$.

The differential ${\pi }_{\ast }$ acts as

$$\begin{array}{cc}\hfill {\pi }_{\ast }\left({\partial }_{x_1}\right)& ={\partial }_{x_1}+2ϵx_1{\partial }_{x_2}-2ϵy_1{\partial }_{y_2},\hfill \\ \hfill {\pi }_{\ast }\left({\partial }_{y_1}\right)& ={\partial }_{y_1}+2ϵy_1{\partial }_{x_2}+2ϵx_1{\partial }_{y_2},\hfill \\ \hfill {\pi }_{\ast }\left({\partial }_{x_2}\right)& ={\partial }_{x_2},{\pi }_{\ast }\left({\partial }_{y_2}\right)={\partial }_{y_2},\hfill \\ \hfill {\pi }_{\ast }\left({\partial }_{x_3}\right)& =0,{\pi }_{\ast }\left({\partial }_{y_3}\right)=0.\hfill \end{array}$$

Thus, $ker{\pi }_{\ast }=\mathrm{span}\{{\partial }_{x_3},{\partial }_{y_3}\}$ and $D^{\perp }\subset ker{\pi }_{\ast }$.

To verify that $h(D,D^{\perp })\ne 0$, take $X=X_1\in D$ and $V={\partial }_{y_3}-{\displaystyle \frac{1}{2x_1+x_2}}{\partial }_{x_1}\in D^{\perp }$ (adjusted for tangency). Compute using the flat connection

$$\begin{array}{cc}\hfill {\overline{\nabla }}_{X}V& ={\overline{\nabla }}_{{\partial }_{x_1}+(2x_1+x_2){\partial }_{x_3}}\left({\partial }_{y_3}-{\displaystyle \frac{1}{2x_1+x_2}}{\partial }_{x_1}\right)\hfill \\ \hfill & ={\displaystyle \frac{2}{{(2x_1+x_2)}^2}}{\partial }_{x_1}-{\partial }_{x_1}=\left({\displaystyle \frac{2}{{(2x_1+x_2)}^2}}-1\right){\partial }_{x_1}.\hfill \end{array}$$

The normal component is

$$h(X,V)=〈\left({\displaystyle \frac{2}{{(2x_1+x_2)}^2}}-1\right){\partial }_{x_1},N〉N,$$

where $\langle {\partial }_{x_1},N\rangle ={\displaystyle \frac{-(2x_1+x_2)}{\parallel \nabla f\parallel }}\ne 0$, so $h(X,V)\ne 0$.

Take horizontal vectors $U={\partial }_{x_1}$, $V={\partial }_{y_1}$. Then

$${\pi }_{\ast }U={\partial }_{u_1}+2\epsilon x_1{\partial }_{u_2}+2\epsilon y_1{\partial }_{v_2},$$

$${\pi }_{\ast }V={\partial }_{v_1}-2\epsilon y_1{\partial }_{u_2}+2\epsilon x_1{\partial }_{v_2}.$$

The coefficient of ${\partial }_{u_2}$ in ${\pi }_{\ast }V$ is $-2\epsilon y_1=-2\epsilon v_1$. The directional derivative along ${\pi }_{\ast }U$ is

$${\partial }_{u_1}(-2\epsilon v_1)+2\epsilon x_1·0+2\epsilon y_1·(-2\epsilon )=-4{\epsilon }^2{y}_1.$$

At normal coordinates where ${\nabla }_U^{M}V=0$, we get

$$(\nabla {\pi }_{\ast })(U,V)=-4{\epsilon }^2{y}_1{\partial }_{u_2}+\mathrm{other}\mathrm{terms}\ne 0$$

whenever $y_1\ne 0$.

Thus, the second fundamental form of the Riemannian map $(\nabla {\pi }_{\ast })$ is non-zero.

This example demonstrates a Riemannian map from a non-totally geodesic generic submanifold where both $h(D,D^{\perp })\ne 0$ and $(\nabla {\pi }_{\ast })\ne 0$ contribute non-trivially to the curvature relations in Theorem 3.

Example 2.

Let $(\overline{M},\overline{J},\overline{g})=({\mathbb{C}}^3,J_0,g_0)$ be the standard flat Kähler manifold with coordinates

$$(z_1,z_2,z_3)=(x_1+i{y}_1,x_2+i{y}_2,x_3+i{y}_3),$$

where

$$J_0\left({\partial }_{x_k}\right)={\partial }_{y_k},J_0\left({\partial }_{y_k}\right)=-{\partial }_{x_k},k=1,2,3,$$

and $g_0=d{x}_1^2+d{y}_1^2+d{x}_2^2+d{y}_2^2+d{x}_3^2+d{y}_3^2$.

Consider the real submanifold

$$M=\{(z_1,z_2,z_3)\in {\mathbb{C}}^3|\Im \left(z_3\right)=|z_1{|}^2,\Re \left(z_3\right)=0\}.$$

Since M has dimension 4, a global coordinate system on M is $(x_1,y_1,x_2,y_2)$.

The basis of $T_{p}M$ (as vectors in $T\overline{M}$) is given by

$$\begin{array}{cc}\hfill F_1& ={\partial }_{x_1}+2x_1{\partial }_{y_3},\hfill \\ \hfill F_2& ={\partial }_{y_1}+2y_1{\partial }_{y_3},\hfill \\ \hfill F_3& ={\partial }_{x_2},\hfill \\ \hfill F_4& ={\partial }_{y_2}.\hfill \end{array}$$

The complex structure $J_0$ acts on them as

$$\begin{array}{cc}\hfill J_0{F}_1& ={\partial }_{y_1}+2x_1{\partial }_{x_3},J_0{F}_2=-{\partial }_{x_1}+2y_1{\partial }_{x_3},\hfill \\ \hfill J_0{F}_3& ={\partial }_{y_2},J_0{F}_4=-{\partial }_{x_2}.\hfill \end{array}$$

One can easily check that the holomorphic distribution

$$D:=T_{p}M\cap J_0\left(T_{p}M\right)=\mathrm{span}\{F_3,F_4\}.$$

The orthogonal complement $D^{\perp }$ of D in $TM$ is

$$D^{\perp }=\mathrm{span}\{F_1,F_2\}.$$

Since $J_0{F}_1$ and $J_0{F}_2$ are not tangent to M, $D^{\perp }$ is a purely real distribution. Thus

$$TM=D\oplus D^{\perp },dimD=2,dim{D}^{\perp }=2,$$

and M is a generic submanifold of $\overline{M}$.

Let us consider $B={\mathbb{C}}^2$ with coordinates $(w_1,w_2)=(u_1+i{v}_1,u_2+i{v}_2)$, equipped with the standard Hermitian structure $(J_B,g_B)$,

$$g_B=d{u}_1^2+d{v}_1^2+d{u}_2^2+d{v}_2^2,J_B\left({\partial }_{u_k}\right)={\partial }_{v_k},J_B\left({\partial }_{v_k}\right)=-{\partial }_{u_k},k=1,2.$$

Define

$$\pi (z_1,z_2,z_3)=\left({\displaystyle \frac{z_2}{\sqrt{2}}},{\displaystyle \frac{z_2}{\sqrt{2}}}\right).$$

Hence

$$ker{\pi }_{\ast }=\mathrm{span}\{F_1,F_2\}=D^{\perp },{(ker{\pi }_{\ast })}^{\perp }=\mathrm{span}\{F_3,F_4\}=D.$$

For $X=a{F}_3+b{F}_4\in D$,

$${\pi }_{\ast }X={\displaystyle \frac{a}{\sqrt{2}}}({\partial }_{u_1}+{\partial }_{u_2})+{\displaystyle \frac{b}{\sqrt{2}}}({\partial }_{v_1}+{\partial }_{v_2}).$$

As $\{{\partial }_{u_1}+{\partial }_{u_2},{\partial }_{v_1}+{\partial }_{v_2}\}$ are orthonormal in $g_B$,

$$g_B({\pi }_{\ast }X,{\pi }_{\ast }X)=a^2+b^2=g_M(X,X),$$

since $F_3,F_4$ are orthonormal in the induced metric $g_M$. Therefore ${\pi }_{\ast }{|}_{{(ker{\pi }_{\ast })}^{\perp }}$ is a linear isometry onto $\mathrm{range}{\pi }_{\ast }$.

On D we have $J_0{F}_3=F_4$ and $J_0{F}_4=-F_3$. Consequently

$${\pi }_{\ast }\left(J_0{F}_3\right)={\pi }_{\ast }\left(F_4\right)={\displaystyle \frac{1}{\sqrt{2}}}({\partial }_{v_1}+{\partial }_{v_2})=J_B\left({\displaystyle \frac{1}{\sqrt{2}}}({\partial }_{u_1}+{\partial }_{u_2})\right)=J_B\left({\pi }_{\ast }{F}_3\right),$$

and similarly ${\pi }_{\ast }\left(J_0{F}_4\right)=J_B\left({\pi }_{\ast }{F}_4\right)$. Hence ${\pi }_{\ast }{|}_D$ is holomorphic.

Thus, π is a proper Riemannian map from a generic submanifold M of a Kähler manifold $\overline{M}={\mathbb{C}}^3$ to an almost Hermitian manifold $B={\mathbb{C}}^2$.

6. Conclusions and Future Research Work

This paper establishes a comprehensive framework for studying Riemannian maps from generic submanifolds of Kähler manifolds, revealing distinctive geometric features that differentiate this setting from the previously studied CR case. The presence of the O’Neill-type tensor C for the generic structure introduces new phenomena not observed in CR geometry.

Our principal contributions include the fundamental relation between the second fundamental form and O’Neill tensors in Lemma 3, the characterization of Kähler structure inheritance on the image distribution in Theorem 1, the harmonicity criterion connecting map harmonicity to fiber minimality in Theorem 2, and the holomorphic sectional curvature relation with its generic correction term in Theorem 3. The explicit examples provided validate the consistency and demonstrate the non-trivial nature of these geometric relations.

Future research may extend this theory to other classes of submanifolds and different ambient manifold structures. Additionally, the various specialized types of Riemannian maps such as those studied for Ricci solitons [22], warped products [26], and Clairaut conditions [24] provide natural directions for future work. Extending the results of this paper and those in [29] to these broader classes of Riemannian maps represents a logical next step.