An odd
\((2m+1)\)-dimensional smooth manifold
M̃ is called a
Kenmotsu manifold, if it is consisting in an endomorphism
φ of its tangent bundle
TM̃, a structure vector field
ξ, and a 1-form
η satisfying the following:
$$\begin{aligned} &\varphi^{2}=-I+\eta\oplus\xi, \qquad \eta(\xi)=1,\qquad \eta o\varphi=0, \end{aligned}$$
(2.1)
$$\begin{aligned} &g(\varphi U, \varphi V)=g(U, V)-\eta(U)\eta(V),\qquad \eta(U)=g(U, \xi), \end{aligned}$$
(2.2)
and the structure equation is given by
$$\begin{aligned} &(\widetilde{\nabla}_{U}\varphi)V=g(\varphi U, V)-\eta(V) \varphi U, \end{aligned}$$
(2.3)
$$\begin{aligned} &\widetilde{\nabla}_{U}\xi=U-\eta(U)\xi, \end{aligned}$$
(2.4)
for any
\(U, V\) tangent to
M̃ (see [
23]). The curvature tensor
R̃ for Kenmotsu space forms is defined as
$$\begin{aligned} \widetilde{R}(X, Y, Z, W)={}&\frac{c-3}{4}\bigl\{ g(X, W)g(Y, Z)-g(X, Z)g(Y, W)\bigr\} \\ &{}+\frac{c+1}{4}\bigl\{ g(X, \varphi W)g(Y, \varphi Z)-g(X, \varphi Z)g(Y, \varphi W) \\ &{}-2g(X, \varphi Y)g(Z, \varphi W)-g(X, W)\eta(Y)\eta(Z) \\ &{}+g(X, Z)\eta(X)\eta(W)-g(Y, Z)\eta(X)\eta(W) \\ &{}+g(Y, W)\eta(X)\eta(Z)\bigr\} , \end{aligned}$$
(2.5)
where
c is a function of the constant
φ-sectional curvature of
M̃ (see [
3].
Let
M be a submanifold of an almost contact metric manifold
M̃ with induced metric
g; if ∇ and
\(\nabla ^{\perp}\) are the induced connections on the tangent bundle
TM and the normal bundle
\(T^{\perp}M\) of
M, respectively, then the Gauss and Weingarten formulas are given by
$$\begin{aligned} &\widetilde{\nabla}_{U}V=\nabla_{U}V+h(U,V), \end{aligned}$$
(2.6)
$$\begin{aligned} &\widetilde{\nabla}_{U}N=-A_{N}U+ \nabla^{\perp}_{U}N, \end{aligned}$$
(2.7)
for each
\(U, V\in\Gamma(TM)\) and
\(N\in\Gamma(T^{\perp}M)\), where
h and
\(A_{N}\) are the second fundamental form and the shape operator (corresponding to the normal vector field
N), respectively, for the immersion of
M into
M̃. They are related as
$$\begin{aligned} g\bigl(h(U, V), N\bigr)=g(A_{N}U, V), \end{aligned}$$
(2.8)
where
g denotes the Riemannian metric on
M̃ as well as the metric induced on
M. Moreover, for a submanifold
M, the Gauss equation is defined as
$$\begin{aligned} \widetilde{R}(U, V, Z, W)={}&R(U, V, Z, W)+g\bigl(h(U, Z), h(V, W) \bigr) \\ &{}-g\bigl(h(U, W), h(V, Z)\bigr), \end{aligned}$$
(2.9)
for any
\(U, V, Z, W\in\Gamma(T M)\), where
R̃ and
R are the curvature tensors on
M̃ and
M, respectively. The mean curvature vector
H for an orthonormal frame
\(\{e_{1}, e_{2},\ldots, e_{n}\}\) of the tangent space
TM on
M is defined by
$$\begin{aligned} H=\frac{1}{n} \operatorname{trace}(h)=\frac{1}{n}\sum _{i=1}^{n}h(e_{i}, e_{i}), \end{aligned}$$
(2.10)
where
\(n=\dim M\). In addition, we set
$$\begin{aligned} h^{r}_{ij}=g\bigl(h(e_{i}, e_{j}), e_{r}\bigr) \quad \mbox{and}\quad \|P \|^{2}=\sum_{i, j=1}^{n} g^{2}(\varphi e_{i}, e_{j}). \end{aligned}$$
(2.11)
Furthermore, the scalar curvature
ρ for a submanifold
M of an almost contact manifold
M̃ is given by
$$\begin{aligned} \rho=\sum_{1\leq i\neq j\leq n}K(e_{i} \wedge e_{j}), \end{aligned}$$
(2.12)
where
\(K(e_{i}\wedge e_{j})\) is the sectional curvature of plane section spanned by
\(e_{i}\) and
\(e_{j}\). Let
\(G_{r}\) be a
r-plane section on
TM and
\(\{e_{1}, e_{2},\ldots, e_{r}\}\) any orthonormal basis of
\(G_{r}\). Then the scalar curvature
\(\rho(G_{r})\) of
\(G_{r}\) is given by
$$\begin{aligned} \rho(G_{r})=\sum_{1\leq i\neq j\leq r}K(e_{i} \wedge e_{j}). \end{aligned}$$
(2.13)
Let
M̃ be a Kenmotsu manifold with an almost contact structure
\((\varphi, \xi, \eta)\) and
M be a submanifold tangent to the structure vector field
ξ isometrically immersed in
M̃. Then
M is called invariant if
\(\varphi(T_{p}M)\subseteq T_{p} M\), and
M is called anti-invariant if
\(\varphi(T_{p}M)\subset T^{\perp}_{p}M\) for every
\(p\in M\) where
\(T_{p}M\) denotes the tangent bundle of
M at the point
p. Moreover,
M is called a slant submanifold if all non-zero vectors
U tangent to
M at a point
p, the angle of
\(\theta(U)\) between
φU and
\(T_{p}M\) are constant,
i.e., they do not depend on the choice of
\(p\in M\) and
\(U\in\Gamma(T_{p}M-\langle\xi (p)\rangle)\) (see [
24]). Except invariant, anti-invariant, and slant submanifolds, there are several other classes of submanifolds determined by the behavior of the tangent space of the submanifold under the action of a one-one tensor field
φ of an ambient manifold.
Assume that
\(\phi:M=M_{1}\times_{f}M_{2}\rightarrow\widetilde{M}\) is an isometric immersion of a warped product
\(M_{1}\times_{f}M_{2}\) into a Riemannian manifold of
M̃ of constant section curvature
c. Suppose that
\(n_{1}, n_{2}\), and
n are the dimensions of
\(M_{1}, M_{2}\), and
\(M_{1}\times_{f}M_{2}\), respectively. Then for unit vector fields
\(X, Z\) tangent to
\(M_{1}, M_{2}\), respectively,
$$\begin{aligned} K(X\wedge Z)&=g(\nabla_{Z}\nabla_{X}X- \nabla_{X}\nabla_{Z}X, Z) \\ &=\frac{1}{f}\bigl\{ (\nabla_{X}X)f-X^{2}f\bigr\} . \end{aligned}$$
(2.17)
Let us assume a local orthonormal frame
\(\{e_{1}, e_{2},\ldots, e_{n}\}\) such that
\(e_{1}, e_{2},\ldots, e_{n_{1}}\) tangent to
\(M_{1}\) and
\(e_{n_{1}+1},\ldots, e_{n}\) are tangent to
\(M_{2}\). Then
$$\begin{aligned} \sum_{1\leq i\leq n_{1}}\sum _{n_{1}+1\leq j\leq n}K(e_{i}\wedge e_{j})= \frac {n_{2}\cdot\Delta f}{f}. \end{aligned}$$
(2.18)