Weak Nearly $\mathcal{S}$- and Weak Nearly $\mathcal{C}$-Manifolds

Abstract

The recent interest in geometers in the f-structures of K. Yano is motivated by the study of the dynamics of contact foliations, as well as their applications in theoretical physics. Weak metric f-structures on a smooth manifold, recently introduced by the author and R. Wolak, open a new perspective on the theory of classical structures. In this paper, we define structures of this kind, called weak nearly $\mathcal{S}$- and weak nearly $\mathcal{C}$-structures, study their geometry, e.g., their relations to Killing vector fields, and characterize weak nearly $\mathcal{S}$- and weak nearly $\mathcal{C}$-submanifolds in a weak nearly Kähler manifold.

1. Introduction

The f-structure introduced by K. Yano [1] on a smooth manifold $M^{2n+s}$ serves as a higher-dimensional analog of almost complex structures ($s=0$) and almost contact structures ($s=1$). This structure is defined by a (1,1)-tensor f of rank $2n$ such that $f^3+f=0$. The tangent bundle splits into two complementary subbundles: $TM=f(TM)\oplus kerf$. The restriction of f to the $2n$-dimensional distribution $f(TM)$ defines a complex structure. The existence of the f-structure on $M^{2n+s}$ is equivalent to a reduction of the structure group to $U(n)\times O(s)$; see [2]. A submanifold M of an almost complex manifold $(\overline{M},J)$ that satisfies the condition $dim(T_{x}M\cap J(T_{x}M))=const>0$ naturally possesses an f-structure; see [3]. An f-structure is a special case of an almost product structure, defined by two complementary orthogonal distributions of a Riemannian manifold $(M,g)$. Foliations appear when one or both distributions are involutive. An interesting case occurs when the sub-bundle $kerf$ is parallelizable, leading to a framed f-structure for which the reduced structure group is $U(n)\times {\mathrm{Id}}_s$. In this scenario, there exist vector fields ${\left\{{\xi }_i\right\}}_{1\le i\le s}$ (called Reeb vector fields) spanning $kerf$ with dual 1-forms ${\left\{{\eta }^i\right\}}_{1\le i\le s}$, satisfying $f^2=-\mathrm{Id}+{\sum }_{i=1}^s{\eta }^i\otimes {\xi }_i$. Compatible Riemannian metrics, i.e.,

$$g(fX,fY)=g(X,Y)-{\sum }_{i=1}^s{\eta }^i(X){\eta }^i(Y),$$

exist on any framed f-manifold, and we obtain the metric f-structure; see [2,4,5,6].

To generalize concepts and results from almost contact geometry to metric f-manifolds, geometers have introduced and studied various broad classes of metric f-structures. A metric f-manifold is termed a $\mathcal{K}$-manifold if it is normal and $d\Phi =0$, where $\Phi (X,Y):=g(X,fY)$. Two important subclasses of $\mathcal{K}$-manifolds are $\mathcal{C}$-manifolds if $d{\eta }^i=0$ and $\mathcal{S}$-manifolds if $d{\eta }^i=\Phi$ for any i; see [2]. Omitting the normality condition, we obtain almost $\mathcal{K}$-manifolds, almost $\mathcal{S}$-manifolds and almost $\mathcal{C}$-manifolds, e.g., [7,8,9]. The distribution $kerf$ of a $\mathcal{K}$-manifold is tangent to a $\mathfrak{g}$-foliation with flat totally geodesic leaves. An f-K-contact manifold is an almost $\mathcal{S}$-manifold, whose Reeb vector fields are Killing vector fields; the structure is intermediate between almost $\mathcal{S}$-structure and S-structure; see [6,10]. Nearly $\mathcal{S}$- and nearly $\mathcal{C}$-manifolds $(M^{2n+s},f,{\xi }_i,{\eta }^i,g)$ are defined in the same spirit as the nearly Kähler manifolds of A. Gray [11] by a constraint only on the symmetric part of $\nabla f$ – starting from $\mathcal{S}$- and $\mathcal{C}$-manifolds (e.g., [12,13,14,15]):

$$({\nabla }_{X}f)X=\left\{\begin{array}{cc}g(fX,fX)\overline{\xi }+\overline{\eta }(X)f^{2}X,& \mathrm{nearly}\mathcal{S}-\mathrm{manifolds}.\\ 0,& \mathrm{nearly}\mathcal{C}-\mathrm{manifolds}.\end{array}\right.$$

Here, $\overline{\eta }={\sum }_{i=1}^s{\eta }^i$ and $\overline{\xi }={\sum }_{i=1}^s{\xi }_i$. These counterparts of nearly Kähler manifolds play a key role in the classification of metric f-manifolds; see [2]. The Reeb vector fields ${\xi }_i$ of nearly $\mathcal{S}$- and nearly $\mathcal{C}$-structures are unit Killing vector fields. The influence of constant-length Killing vector fields on Riemannian geometry has been studied by many authors, e.g., [16]. The interest of geometers in f-structures is also motivated by the study of the dynamics of contact foliations. Contact foliations generalize to higher dimensions the flow of the Reeb vector field on contact manifolds, and $\mathcal{K}$-structures are a particular case of uniform s-contact structures; see [17,18]. Dynamics and integration on s-cosymplectic manifolds are studied in [19]; they investigate the Lie integrability of s-evolution systems in this setting, and develop a Hamilton–Jacobi theory tailored to multi-time Hamiltonian systems, both via symplectification techniques.

In [20,21,22], we introduced and studied metric structures on a smooth manifold, see Definition 1, which generalize almost Hermitian, almost contact (e.g., Sasakian and cosymplectic) and f-structures. Such so-called “weak” structures (the complex structure on the contact distribution is replaced by a nonsingular skew-symmetric tensor) allow us a new look at the theory of classical structures and find new applications. A. Einstein worked on various variants of Unified Field Theory, more recently known as Non-symmetric Gravitational Theory (NGT), see [23]. In this theory, the symmetric part g of the basic tensor $G=g+F$ is associated with gravity, and the skew-symmetric one F is associated with electromagnetism. The theory of weak metric structures is fully consistent with the skew-symmetric part of G; thus, it provides new tools for studying NGT. S. Ivanov and M. Zlatanović developed NGT with linear connections of totally skew-symmetric torsion and gave examples with the skew-symmetric part F of the tensor G obtained using an almost contact metric structure; see [24]. In [25], the author and M. Zlatanović were the first to apply weak metric structures to NGT of totally skew-symmetric torsion with tensor $F(X,Y)=g(X,fY)$ of constant rank.

In this paper, we define and study new structures of this kind, generalizing nearly $\mathcal{S}$- and nearly $\mathcal{C}$-structures. Section 2, following the Introduction, recalls some results regarding weak nearly Kähler manifolds (generalizing nearly Kähler manifolds) and weak metric f-manifolds. Section 3 introduces weak nearly $\mathcal{S}$- and weak nearly $\mathcal{C}$-structures and studies their geometry. Section 4 characterizes weak nearly $\mathcal{C}$- and weak nearly $\mathcal{S}$-submanifolds in weak nearly Kähler manifolds and proves that a weak nearly $\mathcal{C}$-manifold with parallel Reeb vector fields is locally the Riemannian product of a Euclidean space and a weak nearly Kähler manifold. The proofs use the properties of new tensors, as well as classical constructions.

2. Preliminaries

Here, we review some results; see [20,21,22]. Nearly Kähler manifolds $(M,J,g)$ were defined by A. Gray [11] using the condition that only the symmetric part of $\nabla J$ vanishes, where ∇ is the Levi-Civita connection, in contrast to the Kähler case, where $\nabla J=0$. Several authors studied the problem of finding and classifying parallel skew-symmetric 2-tensors (other than almost-complex structures) on a Riemannian manifold, e.g., [26].

Definition 1.

A Riemannian manifold $(M,g)$ of even dimension equipped with a skew-symmetric (1,1)-tensor f such that the tensor $f^2$ is negative-definite is called a weak Hermitian manifold. Such $(M,f,g)$ is called a weak Kähler manifold if $\nabla f=0$. A weak Hermitian manifold is called a weak nearly Kähler manifold if

$$({\nabla }_{X}f)Y+({\nabla }_{Y}f)X=0(X,Y\in {\mathfrak{X}}_M).$$

(1)

A weak metric f-structure on a smooth manifold $M^{2n+s}$$(n,s>0)$ is a set $(f,Q,{\xi }_i,{\eta }^i,g)$, where f is a skew-symmetric $(1,1)$-tensor of rank $2n$, Q is a self-adjoint nonsingular $(1,1)$-tensor, ${\xi }_i(1\le i\le s)$ are orthonormal vector fields, ${\eta }^i$ are dual 1-forms, and g is a Riemannian metric on M, satisfying

$$\begin{array}{c}\hfill f^2=-Q+{\sum }_{i=1}^s{\eta }^i\otimes {\xi }_i, {\eta }^i({\xi }_j)={\delta }_j^i, Q{\xi }_i={\xi }_i,\end{array}$$

(2)

$$\begin{array}{c}\hfill g(fX,fY)=g(X,QY)-{\sum }_{i=1}^s{\eta }^i(X){\eta }^i(Y) (X,Y\in {\mathfrak{X}}_M).\end{array}$$

(3)

In this case, $(M^{2n+s},f,Q,{\xi }_i,{\eta }^i,g)$ is called a weak metric f-manifold.

The geometric meaning of (1) is the same as in the classical case: geodesics are f-planar curves. A curve $\gamma$ is f-planar if the section $\dot{\gamma }\wedge f\dot{\gamma }$ is parallel along the curve. A framed weak f-manifold (i.e., only (2) holds) admits a compatible metric (i.e., also (3) holds) if f in (2) has a skew-symmetric representation, i.e., for any $x\in M$ there exists a frame $\left\{e_i\right\}$ on a neighborhood $U_x\subset M$, for which f has a skew-symmetric matrix.

Example 1.

Take $k>1$ almost Hermitian manifolds $(M_j,f_j,g_j)$. The Riemannian product ${\prod }_{j=1}^k(M_j,{\lambda }_j^{1/2}{f}_j,g_j)$, where ${\lambda }_j>0$ are different constants, is a weak almost Hermitian manifold with $Q={⨁}_j{\lambda }_j{\mathrm{Id}}_j$. We call ${\prod }_j(M_j,{\lambda }_j^{1/2}{f}_j,g_j)$ a $({\lambda }_1,\dots ,{\lambda }_k)$-weighed product of almost Hermitian manifolds $(M_j,f_j,g_j)$; see [27]. The $({\lambda }_1,\dots ,{\lambda }_k)$-weighed product of (nearly) Kähler manifolds is a weak (nearly) Kähler manifold. A nearly Kähler manifold of dimension $\le 4$ is a Kähler manifold; see [11]. The unit sphere $S^6$ in the set of purely imaginary Cayley numbers admits a strictly nearly Kähler structure. The classification of weak nearly Kähler manifolds in dimensions $\ge 4$ is an open problem. The $({\lambda }_1,{\lambda }_2)$-weighed products of 2-dimensional Kähler manifolds are 4-dimensional weak nearly Kähler manifolds. The $({\lambda }_1,{\lambda }_2,{\lambda }_3)$-weighed products of 2-dimensional Kähler manifolds and $({\lambda }_1,{\lambda }_2)$-weighed products of 2- and 4-dimensional Kähler manifolds are 6-dimensional weak nearly Kähler manifolds, and similarly for dimensions $>6$.

Putting $Y={\xi }_j$ in (3), and using ${\eta }^i({\xi }_j)={\delta }_j^i$, we get

$$\begin{array}{c}\hfill {\eta }^j(X)=g(X,{\xi }_j);\end{array}$$

(4)

thus, ${\xi }_j$ is orthogonal to the distribution $\mathcal{D}={\bigcap }_{i=1}^{s}ker{\eta }^i$. For a more intuitive understanding of the role of Q in the f-structure, we explain the following properties:

$$f{\xi }_i=0, {\eta }^i\circ f=0, {\eta }^i\circ Q={\eta }^i, [Q,f]=0.$$

By (2), $f^2{\xi }_i=0$ is true. From this and (2), we get $f^3+fQ=0$. By this, $Q{\xi }_i={\xi }_i$ and $f^2{\xi }_i=0$ we get $0=-f^3{\xi }_i=fQ{\xi }_i=f{\xi }_i$. By $f{\xi }_i=0$, (4), and the skew-symmetry of f, we get ${\eta }^i(fX)=g(fX,{\xi }_i)=-g(X,f{\xi }_i)=0$. From this and condition $\mathrm{rank}f=2n$, we conclude that f the distribution $\mathcal{D}$ of a weak metric f-structure is f-invariant, $\mathcal{D}=f(TM)$ and $dim\mathcal{D}=2n$. By this and $f^3+fQ=0$, we get $f^{3}X=f^2(fX)=-QfX$; hence, $f^3+Qf=0$. This and $f^3+fQ=0$ yield $fQ=Qf$. By symmetry of Q and $Q{\xi }_i={\xi }_i$, we get ${\eta }^i(QX)=g(QX,{\xi }_i)=g(X,Q{\xi }_i)=g(X,{\xi }_i)={\eta }^i(X)$.

Therefore, $TM$ splits as complementary orthogonal sum of $\mathcal{D}$ and $kerf$. A weak metric f-structure $(f,Q,{\xi }_i,{\eta }^i,g)$ is said to be normal if the following tensor is zero:

$$\begin{array}{c}\hfill {\mathcal{N}}^{(1)}(X,Y)=[f,f](X,Y)+2{\sum }_{i=1}^{s}d{\eta }^i(X,Y){\xi }_i(X,Y\in {\mathfrak{X}}_M).\end{array}$$

The Nijenhuis torsion of a (1,1)-tensor S and the derivative of a 1-form $\omega$ are given by

$$\begin{array}{cc}\hfill & [S,S](X,Y)=S^2[X,Y]+[SX,SY]-S[SX,Y]-S[X,SY] (X,Y\in {\mathfrak{X}}_M),\hfill \\ \hfill & d\omega (X,Y)=(1/2)\{X(\omega (Y))-Y(\omega (X))-\omega ([X,Y])\} (X,Y\in {\mathfrak{X}}_M).\hfill \end{array}$$

Using the Levi-Civita connection ∇ of g, one can rewrite $[S,S]$ as

$$\begin{array}{c}\hfill [S,S](X,Y)=(S{\nabla }_{Y}S-{\nabla }_{SY}S)X-(S{\nabla }_{X}S-{\nabla }_{SX}S)Y.\end{array}$$

(5)

The fundamental 2-form $\Phi$ on $(M^{2n+s},f,Q,{\xi }_i,{\eta }^i,g)$ is defined by

$$\Phi (X,Y)=g(X,fY)(X,Y\in {\mathfrak{X}}_M).$$

Proposition 1.

A weak metric f-structure with condition ${\mathcal{N}}^{(1)}=0$ satisfies

$$\begin{array}{l}{\pounds }_{{\xi }_i}f=d{\eta }^j({\xi }_i,·)=0,\\ d{\eta }^i(fX,Y)-d{\eta }^i(fY,X)={\displaystyle \frac{1}{2}}{\eta }^i([\tilde{Q}X,fY]),\\ {\nabla }_{{\xi }_i}{\xi }_j\in \mathcal{D},[X,{\xi }_i]\in \mathcal{D}(1\le i,j\le s,X\in \mathcal{D}).\end{array}$$

Moreover, ${\nabla }_{{\xi }_i}{\xi }_j+{\nabla }_{{\xi }_j}{\xi }_i=0$, that is, $kerf$ defines a totally geodesic distribution.

These tensors on a weak metric f-manifold are well known in the classical theory:

$$\begin{array}{cc}\hfill {\mathcal{N}}_i^{(2)}(X,Y)& :=({\pounds }_{fX}{\eta }^i)(Y)-({\pounds }_{fY}{\eta }^i)(X)=2d{\eta }^i(fX,Y)-2d{\eta }^i(fY,X),\hfill \\ \hfill {\mathcal{N}}_i^{(3)}(X)& :=({\pounds }_{{\xi }_i}f)X=[{\xi }_i,fX]-f[{\xi }_i,X],\hfill \\ \hfill {\mathcal{N}}_{ij}^{(4)}(X)& :=({\pounds }_{{\xi }_i}{\eta }^j)(X)={\xi }_i({\eta }^j(X))-{\eta }^j([{\xi }_i,X])=2d{\eta }^j({\xi }_i,X).\hfill \end{array}$$

Example 2.

Let $M^{2n+s}(f,Q,{\xi }_i,{\eta }^i)$ be a weak framed f-manifold. Consider the product manifold $\overline{M}=M^{2n+s}\times {\mathbb{R}}^s$, where ${\mathbb{R}}^s$ is a Euclidean space with a basis ${𝜕}_1,\dots ,{𝜕}_s$, and define tensors J and $\overline{Q}$ on $\overline{M}$ putting $J(X,{\sum }_{i=1}^s{a}^i{𝜕}_i)=(fX-{\sum }_{i=1}^s{a}^i{\xi }_i,{\sum }_j{\eta }^j(X){𝜕}_j)$ and $\overline{Q}(X,{\sum }_{i=1}^s{a}^i{𝜕}_i)=(QX,{\sum }_{i=1}^s{a}^i{𝜕}_i)$ for $a_i\in C^{\infty }(M)$. It can be shown that $J^2=-\overline{Q}$. The tensors ${\mathcal{N}}_i^{(2)},{\mathcal{N}}_i^{(3)},{\mathcal{N}}_{ij}^{(4)}$ appear when we derive the integrability condition $[J,J]=0$ and express the normality condition ${\mathcal{N}}^{(1)}=0$ for $(f,Q,{\xi }_i,{\eta }^i)$.

Define a “small” (1, 1)-tensor $\tilde{Q}:=Q-\mathrm{Id}$ and note that $[\tilde{Q},f]=0$ and ${\eta }^i\circ \tilde{Q}=0$. The following new tensor (vanishing at $\tilde{Q}=0$)

$$\begin{array}{l}{\mathcal{N}}^{(5)}(X,Y,Z):=fZ(g(X,\tilde{Q}Y))-fY(g(X,\tilde{Q}Z))\hfill \\ +g([X,fZ],\tilde{Q}Y)-g([X,fY],\tilde{Q}Z)+g([Y,fZ]-[Z,fY]-f[Y,Z],\tilde{Q}X),\hfill \end{array}$$

which supplements the sequence ${\mathcal{N}}^{(1)},{\mathcal{N}}_i^{(2)},{\mathcal{N}}_i^{(3)},{\mathcal{N}}_{ij}^{(4)}$, is needed to study the weak metric f-structure. We express the covariant derivative of f using a new tensor ${\mathcal{N}}^{(5)}$:

$$\begin{array}{cc}\hfill & 2g(({\nabla }_{X}f)Y,Z)=3d\Phi (X,fY,fZ)-3d\Phi (X,Y,Z)+g({\mathcal{N}}^{(1)}(Y,Z),fX)\hfill \\ \hfill & +{\sum }_{i=1}^s\left({\mathcal{N}}_i^{(2)}(Y,Z){\eta }^i(X)+2d{\eta }^i(fY,X){\eta }^i(Z)-2d{\eta }^i(fZ,X){\eta }^i(Y)\right)+{\mathcal{N}}^{(5)}(X,Y,Z),\hfill \end{array}$$

where the derivative of a 2-form $\Phi$ is given by

$$\begin{array}{cc}\hfill 3d\Phi (X,Y,Z)& =X\Phi (Y,Z)+Y\Phi (Z,X)+Z\Phi (X,Y)\hfill \\ \hfill & -\Phi ([X,Y],Z)-\Phi ([Z,X],Y)-\Phi ([Y,Z],X).\hfill \end{array}$$

Note that the above equality yields

$$\begin{array}{c}\hfill 3d\Phi (X,Y,Z)=({\nabla }_X\Phi )(Y,Z)+({\nabla }_Y\Phi )(Z,X)+({\nabla }_Z\Phi )(X,Y).\end{array}$$

(6)

For particular values of ${\mathcal{N}}^{(5)}$, we get ${\mathcal{N}}^{(5)}({\xi }_i,{\xi }_j,Z)={\mathcal{N}}^{(5)}({\xi }_i,Y,{\xi }_j)=0$ and

$$\begin{array}{cc}\hfill {\mathcal{N}}^{(5)}(X,{\xi }_i,Z)& =-{\mathcal{N}}^{(5)}(X,Z,{\xi }_i)=g({\mathcal{N}}_i^{(3)}(Z),\tilde{Q}X),\hfill \\ \hfill {\mathcal{N}}^{(5)}({\xi }_i,Y,Z)& =g([{\xi }_i,fZ],\tilde{Q}Y)-g([{\xi }_i,fY],\tilde{Q}Z).\hfill \end{array}$$

Definition 2.

A weak metric f-structure is called a weak almost $\mathcal{K}$-structure if $d\Phi =0$. We define its two subclasses as follows:

(i)

A weak almost $\mathcal{C}$-structure if $\Phi$ and ${\eta }^i(1\le i\le s)$ are closed forms;

(ii)

A weak almost $\mathcal{S}$-structure

if the following is valid:

$$\begin{array}{c}\hfill \Phi =d{\eta }^1=\dots =d{\eta }^s(\mathrm{hence},d\Phi =0).\end{array}$$

(7)

Adding the normality condition, we get weak $\mathcal{K}$-, weak $\mathcal{C}$-, and weak $\mathcal{S}$-structures, respectively. A weak f-K-contact structure is a weak almost $\mathcal{S}$-structure, whose structure vector fields ${\xi }_i$ are Killing, i.e., the tensor $({\pounds }_{{\xi }_i}g)(X,Y)=g({\nabla }_Y{\xi }_i,X)+g({\nabla }_X{\xi }_i,Y)$ vanishes. For $s=1$, weak (almost) $\mathcal{C}$- and weak (almost) $\mathcal{S}$-manifolds reduce to weak (almost) cosymplectic manifolds and weak (almost) Sasakian manifolds, respectively.

Remark 1.

The almost $\mathcal{S}$-structure is also called an f-contact structure, e.g., [21]; then, the $\mathcal{S}$-structure can be regarded as a normal f-contact structure.

Example 3.

(i) To construct a weak metric f-structure $(f,Q,{\xi }_i,{\eta }^i,g)$ on the Riemannian product $M=\overline{M}\times {\mathbb{R}}^s$ of a weak almost Hermitian manifold $(\overline{M},\overline{f},\overline{g})$ with $\Omega (X,Y)=\overline{g}(X,\overline{f}Y)$ and a Euclidean space $({\mathbb{R}}^s,d{y}^2)$, we take any point $(x,y)$ of M and set

$${\xi }_i=(0,{𝜕}_{y^i}),{\eta }^i=(0,d{y}^i),f(X,{𝜕}_{y^i})=(\overline{f}X,0),Q(X,{𝜕}_{y^i})=(-{\overline{f}}^{2}X,{𝜕}_{y^i}),$$

where $X\in T_x\overline{M}$. Note that $\nabla f=0$ if and only if $\overline{\nabla }\overline{f}=0$. On the other hand, $\overline{\nabla }\overline{f}=0$ if and only if $d\Omega =0$, see (6) with $\Phi =\Omega$, i.e., $(M,\Omega )$ is a symplectic manifold.

(ii) For a weak $\mathcal{C}$-structure, we obtain $g(({\nabla }_{X}f)Y,Z)={\displaystyle \frac{1}{2}}{\mathcal{N}}^{(5)}(X,Y,Z)$. A weak metric f-structure with conditions $\nabla f=0$ and $g([{\xi }_i,{\xi }_j],{\xi }_k)=0$ is a weak $\mathcal{C}$-structure with the property ${\mathcal{N}}^{(5)}=0$. For a weak $\mathcal{S}$-structure, we get

$$\begin{array}{c}\hfill g(({\nabla }_{X}f)Y,Z)=g(fX,fY)\overline{\eta }(Z)-g(fX,fZ)\overline{\eta }(Y)+{\displaystyle \frac{1}{2}}{\mathcal{N}}^{(5)}(X,Y,Z);\end{array}$$

${\xi }_i$ are Killing vector fields and $kerf$ defines a Riemannian totally geodesic foliation. In particular, for an $\mathcal{S}$-structure, we have

$$\begin{array}{c}\hfill ({\nabla }_{X}f)Y=g(fX,fY)\overline{\xi }+\overline{\eta }(Y)f^{2}X.\end{array}$$

(8)

For a weak almost $\mathcal{K}$-structure (and its special cases, a weak almost $\mathcal{S}$-structure and a weak almost $\mathcal{C}$-structure), the distribution $kerf$ is involutive (tangent to a foliation). Moreover, weak almost $\mathcal{S}$- and weak almost $\mathcal{C}$-structures satisfy the following conditions (trivial for $s=1$):

$$\begin{array}{cc}\hfill [{\xi }_i,{\xi }_j]& =0,\hfill \end{array}$$

(9)

$$\begin{array}{c}g({\nabla }_X{\xi }_i,{\xi }_j)=0(X\in {\mathfrak{X}}_M)\end{array}$$

(10)

for $1\le i,j\le s$. The following condition is a corollary of (10):

$$\begin{array}{c}\hfill {\eta }^k({\nabla }_{{\xi }_i}{\xi }_j)=0(1\le i,j,k\le s).\end{array}$$

(11)

By (9), the distribution $kerf$ of weak almost $\mathcal{S}$- and a weak almost $\mathcal{C}$-manifolds is tangent to a $\mathfrak{g}$-foliation with an abelian Lie algebra.

Remark 2

([28]). Let $\mathfrak{g}$ be a Lie algebra of dimension s. A foliation of dimension s on a smooth connected manifold M is called a $\mathfrak{g}$-foliation if there exist complete vector fields ${\xi }_1,\dots ,{\xi }_s$ on M which, when restricted to each leaf, form a parallelism of this submanifold with a Lie algebra isomorphic to $\mathfrak{g}$.

3. Main Results

In this section, weak nearly $\mathcal{S}$- and weak nearly $\mathcal{C}$-structures are defined and studied; some of the statements generalize the results in [13,14,15].

The restriction on the symmetric part of (8) gives the following.

Definition 3.

A weak metric f-manifold is called a weak nearly $\mathcal{S}$-manifold if

$$({\nabla }_{X}f)Y+({\nabla }_{Y}f)X=2g(fX,fY)\overline{\xi }+\overline{\eta }(X)f^{2}Y+\overline{\eta }(Y)f^{2}X$$

(12)

for all $X,Y\in {\mathfrak{X}}_M$. A weak metric f-manifold is called a weak nearly $\mathcal{C}$-manifold if

$$({\nabla }_{X}f)Y+({\nabla }_{Y}f)X=0.$$

(13)

Example 4.

Let a Riemannian manifold $(M^{2n+s},g)$ admit two nearly $\mathcal{S}$-structures (or, nearly $\mathcal{C}$-structures) $M^{2n+s}(f_k,Q,{\xi }_i,{\eta }^i,g)(k=1,2)$ with common Reeb vector fields ${\xi }_i$ and one-forms ${\eta }^i=g({\xi }_i,·)$. Suppose that $f_1\ne f_2$ are such that $\psi :=f_1{f}_2+f_2{f}_1\ne 0$. Then, $f:=(cost)f_1+(sint)f_2$ for small $t>0$ satisfies (12) (and (13), respectively) and

$$f^2=-\mathrm{Id}+(sintcost)\psi +{\sum }_{i=1}^s{\eta }^i\otimes {\xi }_i.$$

Thus, $(f,Q,{\xi }_i,{\eta }^i,g)$ is a weak nearly $\mathcal{S}$-structure (and weak nearly $\mathcal{C}$-structure, respectively) on $M^{2n+s}$ with $Q=\mathrm{Id}-(sintcost)\psi$.

The following condition is trivial when $Q={\mathrm{Id}}_{TM}$:

$$({\nabla }_{X}Q)Y=0(X,Y\in {\mathfrak{X}}_M,Y\perp kerf).$$

(14)

Using (14), we have

$$\begin{array}{c}\hfill ({\nabla }_{X}Q)Y={\sum }_{i=1}^s{\eta }^i(Y)({\nabla }_{X}Q){\xi }_i=-{\sum }_{i=1}^s{\eta }^i(Y)\tilde{Q}{\nabla }_X{\xi }_i(X,Y\in {\mathfrak{X}}_M).\end{array}$$

Example 5.

To construct a weak (nearly) $\mathcal{C}$-structure $(f,Q,{\xi }_i,{\eta }^i,g)$ on the Riemannian product $M=\overline{M}\times {\mathbb{R}}^s$ of a weak (nearly) Kähler manifold $(\overline{M},\overline{f},\overline{g})$ and a Euclidean space $({\mathbb{R}}^s,d{y}^2)$, we take any point $(x,y)$ of M and set

$${\xi }_i=(0,{𝜕}_{y^i}),{\eta }^i=(0,d{y}^i),f(X,{𝜕}_{y^i})=(\overline{f}X,0),Q(X,{𝜕}_{y^i})=(-{\overline{f}}^{2}X,{𝜕}_{y^i}),$$

as in Example 3(i). Note that if ${\overline{\nabla }}_X({\overline{f}}^2)=0(X\in T\overline{M})$, then (14) holds.

The following result opens new applications to Killing vector fields.

Proposition 2.

Both on a weak nearly $\mathcal{S}$-manifold and a weak nearly $\mathcal{C}$-manifold satisfying (9) and (11), the distribution $kerf$ defines a flat totally geodesic foliation; moreover, if conditions (10) and (14) hold, then the vector fields ${\xi }_i$ are Killing.

Proof. 

Putting $X={\xi }_j$ and $Y={\xi }_k$ in (12) or (13), we find $({\nabla }_{{\xi }_j}f){\xi }_k+({\nabla }_{{\xi }_k}f){\xi }_j=0$; hence, $f\left({\nabla }_{{\xi }_j}{\xi }_k+{\nabla }_{{\xi }_k}{\xi }_j\right)=0$. Applying f to this and using (2), we obtain

$$\begin{array}{c}\hfill 0=f^2\left({\nabla }_{{\xi }_j}{\xi }_k+{\nabla }_{{\xi }_k}{\xi }_j\right)=-Q\left({\nabla }_{{\xi }_j}{\xi }_k+{\nabla }_{{\xi }_k}{\xi }_j\right)+{\sum }_{i=1}^s{\eta }^i\left({\nabla }_{{\xi }_j}{\xi }_k+{\nabla }_{{\xi }_k}{\xi }_j\right){\xi }_i.\end{array}$$

Since the (1,1)-tensor Q is nonsingular and (11) is true, we get ${\nabla }_{{\xi }_j}{\xi }_k+{\nabla }_{{\xi }_k}{\xi }_j=0$. Combining this with ${\nabla }_{{\xi }_j}{\xi }_k-{\nabla }_{{\xi }_k}{\xi }_j=0$, see (9), yields

$$\begin{array}{c}\hfill {\nabla }_{{\xi }_j}{\xi }_k=0(1\le j,k\le s);\end{array}$$

(15)

hence, $kerf$ defines a flat totally geodesic foliation. Next, using (15) we calculate

$$\begin{array}{c}\hfill {\nabla }_{{\xi }_i}{\eta }^j=g({\nabla }_{{\xi }_i}{\xi }_j,·)=0.\end{array}$$

(16)

Using (10) and (15), we obtain

$$({\pounds }_{{\xi }_j}g)({\xi }_k,·)=g({\nabla }_{{\xi }_j}{\xi }_k,·)=0.$$

Taking the ${\xi }_j$-derivative of (3) and using (14) and ${\nabla }_{{\xi }_j}{\eta }^i=0$, we find (for $Y\perp kerf$)

$$\begin{array}{cc}& g(({\nabla }_{{\xi }_j}f)X,fY)+g(fX,({\nabla }_{{\xi }_j}f)Y)={\nabla }_{{\xi }_j}g(fX,fY)\\ & =g(X,({\nabla }_{{\xi }_j}Q)Y)+{\sum }_{i=1}^s\left\{({\nabla }_{{\xi }_j}{\eta }^i)(X){\eta }^i(Y)+{\eta }^i(X)({\nabla }_{{\xi }_j}{\eta }^i)(Y)\right\}=0.\end{array}$$

For a weak nearly $\mathcal{S}$-manifold, using (12), (10), and $\eta \circ \tilde{Q}=0$ yields

$$\begin{array}{lll}& & g(({\nabla }_{{\xi }_j}f)X,fY)+g(fX,({\nabla }_{{\xi }_j}f)Y)\hfill \\ & & =-g(({\nabla }_{X}f){\xi }_j,fY)-g(fX,({\nabla }_{Y}f){\xi }_j)+g(f^{2}X,fY)+g(f^{2}Y,fX)\hfill \\ & & =-g({\nabla }_X{\xi }_j,f^{2}Y)-g(f^{2}X,{\nabla }_Y{\xi }_j)=g({\nabla }_X{\xi }_j,QY)+g(QX,{\nabla }_Y{\xi }_j)\hfill \\ & & =g({\nabla }_X{\xi }_j,Y)+g(X,{\nabla }_Y{\xi }_j)+g({\nabla }_X{\xi }_j,\tilde{Q}Y)+g(\tilde{Q}X,{\nabla }_Y{\xi }_j)\hfill \\ & & =({\pounds }_{{\xi }_j}g)(X,Y)-g({\xi }_j,({\nabla }_X\tilde{Q})Y)-g(({\nabla }_Y\tilde{Q})X,{\xi }_j)=({\pounds }_{{\xi }_j}g)(X,Y).\hfill \end{array}$$

Here, we used $g({\xi }_j,({\nabla }_X\tilde{Q})Y)=0$. For a weak nearly $\mathcal{C}$-manifold, using (13) yields

$$\begin{array}{c}\hfill ({\pounds }_{{\xi }_j}g)(X,Y)=g(({\nabla }_{{\xi }_j}f)X,fY)+g(fX,({\nabla }_{{\xi }_j}f)Y)=0.\end{array}$$

(17)

From (17), for both cases we obtain ${\pounds }_{{\xi }_j}g=0$, i.e., ${\xi }_j$ is a Killing vector field. □

Remark 3.

Note that even for a nearly $\mathcal{S}$-manifold without conditions (9) and (10), the vector fields ${\xi }_i(1\le i\le s)$ are not Killing; see Corollary 1 in [13].

Theorem 1.

There are no weak nearly $\mathcal{C}$-manifolds with conditions (9), (10), and (14) which satisfy $\Phi =d{\eta }^1=\dots =d{\eta }^s$; see (7).

Proof. 

Suppose that our weak nearly $\mathcal{C}$-manifold satisfies (7). Since also ${\xi }_i$ are Killing vector fields (see Proposition 2), M is a weak f-K-contact manifold. By Theorem 1 in [22], the following holds:

$$\begin{array}{c}\hfill \nabla {\xi }_i=-f(1\le i\le s).\end{array}$$

(18)

By Proposition 6 in [22], the $\xi$-sectional curvature of a weak f-K-contact manifold is positive, i.e., $K({\xi }_i,X)>0(X\perp kerf)$. Thus, for any nonzero vector $X\perp kerf$, using (13) and (18), we get

$$\begin{array}{c}0<K({\xi }_i,X)=g({\nabla }_{{\xi }_i}{\nabla }_X{\xi }_i-{\nabla }_X{\nabla }_{{\xi }_i}{\xi }_i-{\nabla }_{[{\xi }_i,X]}{\xi }_i,X)\hfill \\ =g(-({\nabla }_{{\xi }_i}f)X+f^{2}X,X)=g(({\nabla }_{X}f){\xi }_i,X)-g(fX,fX)\hfill \\ =-g(f{\nabla }_X{\xi }_i,X)+g(f^{2}X,X)=2g(f^{2}X,X).\hfill \end{array}$$

This contradicts the following equality: $g(f^{2}X,X)=-g(fX,fX)\le 0$. □

Corollary 1.

There are no nearly $\mathcal{C}$-manifolds with conditions (9) and (10) which satisfy (7).

Theorem 2.

A weak nearly $\mathcal{C}$-manifold $(M^{2n+s},f,Q,{\xi }_i,{\eta }^i,g)$ satisfies

$$\nabla {\xi }_i=0(1\le i\le s)$$

(19)

if and only if the manifold is locally isometric to the Riemannian product of a Euclidean s-space and a weak nearly Kähler manifold.

Proof. 

For all vector fields $X,Y$ orthogonal to $kerf$, we have

$$2d{\eta }^i(X,Y)=g({\nabla }_X{\xi }_i,Y)-g({\nabla }_Y{\xi }_i,X).$$

(20)

Thus, if the condition $\nabla {\xi }_i=0$ holds, then the contact distribution $\mathcal{D}$ is integrable. Moreover, any integral submanifold of $\mathcal{D}$ is a totally geodesic submanifold. Indeed, for $X,Y\perp kerf$, we have $g({\nabla }_{X}Y,{\xi }_i)=-g(Y,{\nabla }_X{\xi }_i)=0$. Since ${\nabla }_{{\xi }_i}{\xi }_j=0$, by de Rham Decomposition Theorem, the manifold is locally the Riemannian product $\overline{M}\times {\mathbb{R}}^s$. The metric weak f-structure induces on $\overline{M}$ a weak almost-Hermitian structure, which, by these conditions, is weak nearly Kähler.

Conversely, if a weak nearly $\mathcal{C}$-manifold is locally the Riemannian product $\overline{M}\times {\mathbb{R}}^s$, where $\overline{M}$ is a weak nearly Kähler manifold and ${\xi }_i=(0,{𝜕}_{y^i})$ (see also Example 5), then $d{\eta }^j(X,Y)=0(X,Y\perp kerf)$. By (20) and ${\nabla }_{{\xi }_i}{\xi }_j=0$, we obtain $\nabla {\xi }_i=0$. □

Corollary 2.

A nearly $\mathcal{C}$-manifold $(M^{2n+s},f,{\xi }_i,{\eta }^i,g)$ satisfies (19) if and only if the manifold is locally isometric to the Riemannian product of ${\mathbb{R}}^s$ and a nearly Kähler manifold.

Theorem 3.

Let a weak nearly $\mathcal{S}$-structure satisfy (9), (10), and (14); then, the following is true:

(i) 

The condition ${\eta }^j\circ N^{(1)}=0(1\le j\le s)$ yields $d{\eta }^j(X,Y)=\Phi (QX,Y)$ for all j.

(ii) 

The condition (7) yields $N^{(1)}(X,Y)=2\Phi (\tilde{Q}X,Y)\overline{\xi }$.

Proof. 

$(i)$ We calculate, using (5), (12), and ${\eta }^j\circ f=0$,

$$\begin{array}{cc}\hfill {\eta }^j(N^{(1)}(X,Y))& -2d{\eta }^j(X,Y)={\eta }^j([f,f](X,Y))\stackrel{(5)}{=}{\eta }^j\left(({\nabla }_{fX}f)Y-({\nabla }_{fY}f)X\right)\hfill \\ \hfill & \stackrel{(12)}{=}{\eta }^j\left(({\nabla }_{X}f)fY-({\nabla }_{Y}f)fX\right)+4g(f^{2}X,fY)\hfill \\ \hfill & =g\left(({\nabla }_X{f}^2)Y-({\nabla }_Y{f}^2)X,{\xi }_j\right)-4g(QX,fY)\hfill \\ \hfill & =({\nabla }_X{\eta }^j)(Y)-({\nabla }_Y{\eta }^j)(X)-4g(QX,fY)\hfill \\ \hfill & =2d{\eta }^j(X,Y)-4g(QX,fY).\hfill \end{array}$$

Here, we used the identity $2d{\eta }^j(X,Y)=({\nabla }_X{\eta }^j)(Y)-({\nabla }_Y{\eta }^j)(X)$.

Thus, if ${\eta }^i(N^{(1)}(X,Y))=0$, then $d{\eta }^j(X,Y)=g(QX,fY)=\Phi (QX,Y)$ for all j.

$(ii)$ Using $d\Phi =0$, (2), (6), and (12), where $\overline{\eta }={\sum }_{i=1}^s{\eta }^i$ and $\overline{\xi }={\sum }_{i=1}^s{\xi }_i$, we get

$$\begin{array}{cc}\hfill & 3d\Phi (X,Y,Z)=-g(({\nabla }_{X}f)Y,Z)+g(({\nabla }_{Y}f)X,Z)-g(({\nabla }_{Z}f)X,Y)\hfill \\ \hfill & =-g(({\nabla }_{X}f)Y,Z)+g\left(-({\nabla }_{X}f)Y+2g(fX,fY)\overline{\xi }+\overline{\eta }(X)f^{2}Y+\overline{\eta }(Y)f^{2}X,Z\right)\hfill \\ \hfill & +g\left(({\nabla }_{X}f)Z-2g(fX,fZ)\overline{\xi }-\overline{\eta }(X)f^{2}Z-\overline{\eta }(Z)f^{2}X,Y\right)\hfill \\ \hfill & =-3g(({\nabla }_{X}f)Y,Z)-3g(f^{2}X,Y)\overline{\eta }(Z)+3g(f^{2}X,Z)\overline{\eta }(Y).\hfill \end{array}$$

Thus, (8) holds. Using (8) in (5) gives

$$[f,f]=2g(f^{2}X,fY)\overline{\xi }=-2g(QX,fY)\overline{\xi }=-2\Phi (QX,Y)\overline{\xi },$$

hence, $N^{(1)}(X,Y)=2\Phi (\tilde{Q}X,Y)\overline{\xi }$. □

A consequence of Theorem 3 is a rigidity result for $\mathcal{S}$-manifolds; see Theorem 1 of [13].

Corollary 3.

A normal nearly $\mathcal{S}$-structure is an $\mathcal{S}$-structure.

4. Submanifolds of Weak Nearly Kähler Manifolds

Here, we study weak nearly $\mathcal{S}$- and weak nearly $\mathcal{C}$- submanifolds in a weak nearly Kähler manifold. The second fundamental form h of a submanifold $M\subset (\overline{M},\overline{g})$ is related with $\overline{\nabla }$ (the Levi-Civita connection of $\overline{g}$ restricted to M) and ∇ (the Levi-Civita connection of metric g induced on M via the Gauss equation) by

$${\overline{\nabla }}_{X}Y={\nabla }_{X}Y+h(X,Y)(X,Y\in {\mathfrak{X}}_M).$$

(21)

A submanifold is said to be totally geodesic if $h=0$. The shape operator $A_N:X\mapsto -{\overline{\nabla }}_{X}N$ with respect to a unit normal N is related with h via the equalities

$$h_N(X,Y)=\overline{g}(h(X,Y),N)=g(A_N(X),Y)(X,Y\in {\mathfrak{X}}_M).$$

(22)

Lemma 1.

Let $(\overline{M},\overline{f},\overline{g})$ be a weak Hermitian manifold and $M^{2n+s}$ a submanifold of codimension s equipped with mutually orthogonal unit normals $N_i(i=1,\dots ,s)$ satisfying the condition

$$\begin{array}{c}\hfill \overline{g}(\overline{f}{N}_i,N_j)=0(1\le i,j\le s)\end{array}$$

(23)

(trivial for $s=1$). Then, M inherits a metric weak f-structure $(f,Q,{\xi }_i,{\eta }^i,g)$ given by

$$\begin{array}{ll}\hfill & {\xi }_i=\overline{f}{N}_i,{\eta }^i=\overline{g}(\overline{f}{N}_i,·)(i=1,\dots ,s),g=\overline{g}{|}_M,\hfill \\ \hfill & f=\overline{f}+{\sum }_{i=1}^s\overline{g}(\overline{f}{N}_i,·)N_i,Q=-{\overline{f}}^2+{\sum }_{i=1}^s\overline{g}({\overline{f}}^2{N}_i,·)N_i.\hfill \end{array}$$

(24)

Moreover, (14) holds on M if ${\overline{f}}^2{N}_i\perp TM(1\le i\le s)$ and

$${(({\overline{\nabla }}_X{\overline{f}}^2)Y)}^{\top }=0(X,Y\in {\mathfrak{X}}_M,Y\perp kerf).$$

Proof. 

Using the skew-symmetry of $\overline{f}$ and (23), we verify (2):

$$\begin{array}{l}{f}^{2}X=f(\overline{f}X-{\sum }_{i=1}^s\overline{g}(\overline{f}X,N_i)N_i)\hfill \\ =\overline{f}\left(\overline{f}X-{\sum }_{i=1}^s\overline{g}(\overline{f}X,N_i)N_i\right)-\overline{g}(\overline{f}(\overline{f}X-{\sum }_{i,j=1}^s\overline{g}(\overline{f}X,N_i)N_i),N_j)N_j\hfill \\ ={\overline{f}}^{2}X-{\sum }_j\overline{g}({\overline{f}}^2{N}_j,X)N_j-{\sum }_{i=1}^s\overline{g}(\overline{f}{N}_i,X)\overline{f}{N}_i+{\sum }_{i,j=1}^s\overline{g}(\overline{f}X,N_i)\overline{g}(\overline{f}{N}_i,N_j)N_j\hfill \\ =-QX+{\sum }_{i=1}^s{\eta }^i(X){\xi }_i(X\in {\mathfrak{X}}_M).\hfill \end{array}$$

Since ${\overline{f}}^2$ is negative-definite, for nonzero $X\in {\mathfrak{X}}_M$ we obtain $\overline{g}(N_i,X)=0$ and

$$g(QX,X)=\overline{g}(-{\overline{f}}^{2}X+{\sum }_{i=1}^s\overline{g}({\overline{f}}^2{N}_i,X)N_i,X)=-\overline{g}({\overline{f}}^{2}X,X)>0,$$

hence, the tensor Q is positive-definite on $TM$. Then, we calculate $({\nabla }_{X}Q)Y$ for $X,Y\in {\mathfrak{X}}_M$ and $Y\perp kerf$, using (21) and (24) and the condition ${\overline{f}}^2{N}_i\perp TM(1\le i\le s)$:

$$\begin{array}{l} ({\nabla }_{X}Q)Y={\nabla }_X(QY)-Q({\nabla }_{X}Y)\hfill \\ =\{{\overline{\nabla }}_X\left(-{\overline{f}}^{2}Y+{\sum }_{i=1}^{s}g({\overline{f}}^2{N}_i,Y)N_i\right)-h(X,QY)+{\overline{f}}^2\left({\overline{\nabla }}_{X}Y-h(X,Y)\right)\hfill \\ -{\sum }_{i=1}^{s}g\left({\overline{f}}^2{N}_i,{\overline{\nabla }}_{X}Y-h(X,Y)\right)N_i{\}}^{\top }\hfill \\ ={(-({\overline{\nabla }}_X({\overline{f}}^{2}Y))+{\overline{f}}^2({\overline{\nabla }}_{X}Y))}^{\top }-{\sum }_{i=1}^{s}g({\overline{f}}^2{N}_i,Y)A_{N_i}X\hfill \\ =-{(({\overline{\nabla }}_X{\overline{f}}^2)Y)}^{\top },\hfill \end{array}$$

where ^⊤ is the $TM$-component of a vector. This completes the proof. □

The following theorem characterizes weak nearly $\mathcal{C}$- and weak nearly $\mathcal{S}$-submanifolds of a nearly Kähler manifold, using the property of the second fundamental form.

Theorem 4.

Let $(\overline{M},\overline{f},\overline{g})$ be a weak nearly Kähler manifold and $M^{2n+s}$ a submanifold of codimension s equipped with mutually orthogonal unit normals $N_i(i=1,\dots ,s)$ satisfying (23). If the second fundamental form of M and the induced metric weak f-structure $(f,Q,{\xi }_i,{\eta }^i,g)$ on M, given by (24), satisfy

$$\begin{array}{l}(i)h_{N_i}(X,Y)=g(QX,Y)+{\sum }_{j,k=1}^s\left(h_{N_i}({\xi }_j,{\xi }_k)-{\delta }_{j,k}\right){\eta }^j(X){\eta }^k(Y),\\ (ii)h_{N_i}(X,Y)={\sum }_{j,k=1}^s{h}_{N_i}({\xi }_j,{\xi }_k){\eta }^j(X){\eta }^k(Y),\end{array}$$

(25)

and

$$\begin{array}{c}{h}_{N_i}({\xi }_j,{\xi }_k)=h_{N_j}({\xi }_i,{\xi }_k)(1\le i,j\le s),\end{array}$$

(26)

then $(f,Q,{\xi }_i,{\eta }^i,g)$ is

$$\begin{array}{c}\hfill (i)a weak nearly \mathcal{S}\text{-}structure;(ii)a weak nearly \mathcal{C}\text{-}structure.\end{array}$$

(27)

Proof. 

Substituting

$$\overline{f}Y=fY-{\sum }_{i=1}^s\overline{g}(\overline{f}{N}_i,Y)N_i=fY-{\sum }_{i=1}^s{\eta }^i(Y)N_i$$

in $({\overline{\nabla }}_X\overline{f})Y$, where $X,Y\in {\mathfrak{X}}_M$, and using (21) and Lemma 1, we obtain

$$\begin{array}{cc}\hfill ({\overline{\nabla }}_X\overline{f})Y={\overline{\nabla }}_X(\overline{f}Y)-\overline{f}({\overline{\nabla }}_{X}Y)& =({\nabla }_{X}f)Y+{\sum }_{i=1}^s\left\{{\eta }^i(Y)A_{N_i}X-h_{N_i}(X,Y){\xi }_i\right\}\hfill \\ \hfill & +{\sum }_{i=1}^s\left\{X({\eta }^i(Y))-{\eta }^i({\nabla }_{X}Y)+h_{N_i}(X,fY)\right\}{N}_i.\hfill \end{array}$$

Thus, the $TM$-component of the weak nearly Kähler condition (1), using (21) and (22), takes the form

$$\begin{array}{c}{\left(({\overline{\nabla }}_X\overline{f})Y+({\overline{\nabla }}_Y\overline{f})X\right)}^{\top }=({\nabla }_{X}f)Y+({\nabla }_{Y}f)X\hfill \\ +{\sum }_{i=1}^s\left\{{\eta }^i(X)A_{N_i}Y+{\eta }^i(Y)A_{N_i}X-2h_{N_i}(X,Y){\xi }_i\right\}=0.\hfill \end{array}$$

(28)

Using (22), one can show that (25) is equivalent to the following:

$$\begin{array}{cc}(i)\hfill & A_{N_i}X=-f^{2}X+{\sum }_{j,k=1}^s{h}_{N_i}({\xi }_j,{\xi }_k){\eta }^j(X){\xi }_k,\hfill \\ (ii)\hfill & A_{N_i}X={\sum }_{j,k=1}^s{h}_{N_i}({\xi }_j,{\xi }_k){\eta }^j(X){\xi }_k.\hfill \end{array}$$

(29)

(i) If we have a weak nearly $\mathcal{S}$-structure, see (12), then from (28) we get

$$\begin{array}{c}2g(fX,fY)\overline{\xi }+\overline{\eta }(Y)f^{2}X+\overline{\eta }(X)f^{2}Y\hfill \\ +{\sum }_{i=1}^s\left\{{\eta }^i(X)A_{N_i}Y+{\eta }^i(Y)A_{N_i}X-2h_{N_i}(X,Y){\xi }_i\right\}=0,\hfill \end{array}$$

(30)

Substituting the expressions of $h_{N_i}(X,Y)$ and $A_{N_i}$, see (25)(i) and (29)(i), in (30) and using (26) gives identity; thus, we obtain a weak nearly $\mathcal{S}$-structure on M.

(ii) If we have a weak nearly $\mathcal{C}$-structure, see (13), then from (28) we get

$$\begin{array}{c}\hfill {\sum }_{i=1}^s\left\{{\eta }^i(X)A_{N_i}Y+{\eta }^i(Y)A_{N_i}X-2h_{N_i}(X,Y){\xi }_i\right\}=0.\end{array}$$

(31)

Substituting the expressions of $h_{N_i}(X,Y)$ and $A_{N_i}$, see (25)(ii) and (29)(ii), in (31) and using (26) gives identity; thus, we obtain a weak nearly $\mathcal{C}$-structure on M. □

For $Q=\mathrm{Id}$, the properties of (25) lead us to the following.

Definition 4.

A codimension s submanifold $M^{2n+s}$ of a Hermitian manifold $(\overline{M},\overline{f},\overline{g})$, equipped with mutually orthogonal unit normals $N_i(i=1,\dots ,s)$ satisfying

$$\begin{array}{c}\hfill h_{N_i}(X,Y)=a_{i}g(X,Y)+{\sum }_{j,k=1}^s{b}_{i,j,k}{\eta }^j(X){\eta }^k(Y),\end{array}$$

(32)

where $a_i,b_{i,j,k}\in C^{\infty }(M)$ and ${\eta }^i(1\le i\le s)$ are linear independent one-forms on M, will be called an s-quasi-umbilical submanifold. For $s=1$, condition (32) reads as follows, see [15]:

$$h_N(X,Y)=a_{1}g(X,Y)+b_1\eta (X)\eta (Y).$$

The geometric meaning of (32) is that the restriction of $h_{N_i}$ on the distribution ${\bigcap }_{i=1}^{s}ker{\eta }^i$ looks similar to h for totally umbilical submanifolds: $h=({\mathrm{trace}}_{g}h/dimM)g$.

The following consequence of Theorem 4 extends the fact (see Theorem 4.1 in [14]) that a hypersurface of a nearly Kähler manifold is nearly Sasakian or nearly cosymplectic if and only if it is quasi-umbilical with respect to the almost contact form.

Corollary 4.

Let $(\overline{M},\overline{f},\overline{g})$ be a nearly Kähler manifold and $M^{2n+s}$ a submanifold of codimension s equipped with mutually orthogonal unit normals $N_i(i=1,\dots ,s)$ satisfying (23), and $(f,{\xi }_i,{\eta }^i,g=\overline{g}{|}_M)$ the induced metric f-structure on M, given by

$$\begin{array}{c}\hfill {\xi }_i=\overline{f}{N}_i,{\eta }^i=\overline{g}(\overline{f}{N}_i,·)(i=1,\dots ,s),f=\overline{f}+{\sum }_{j=1}^s\overline{g}(\overline{f}{N}_j,·)N_j.\end{array}$$

If $M^{2n+s}$ is an s-quasi-umbilical submanifold (with respect to the 1-forms ${\eta }^i$),

$$\begin{array}{l}(i)h_{N_i}(X,Y)=g(X,Y)+{\sum }_{j,k=1}^s\left(h_{N_i}({\xi }_j,{\xi }_k)-{\delta }_{j,k}\right){\eta }^j(X){\eta }^k(Y),\\ (ii)h_{N_i}(X,Y)={\sum }_{j,k=1}^s{h}_{N_i}({\xi }_j,{\xi }_k){\eta }^j(X){\eta }^k(Y),\end{array}$$

and (26) are true, then $(f,{\xi }_i,{\eta }^i,g)$ is (i) a nearly $\mathcal{S}$-structure; (ii) a nearly $\mathcal{C}$-structure.

5. Conclusions

We have shown that weak nearly $\mathcal{S}$- and weak nearly $\mathcal{C}$-structures are useful for studying metric f-structures, e.g., totally geodesic foliations, Killing vector fields, and s-quasi-umbilical submanifolds. Some classical results have been extended in this paper to weak nearly $\mathcal{S}$- and weak nearly $\mathcal{C}$-manifolds with additional conditions. Based on the numerous applications of nearly Kähler, nearly Sasakian, and nearly cosymplectic structures, we expect that weak nearly Kähler, $\mathcal{S}$- and $\mathcal{C}$-structures will be useful for geometry and theoretical physics, e.g., for NGT, the theory of s-cosymplectic structures and s-contact structures, multi-time Hamiltonian systems, and s-evolution systems.