Invariant Submanifolds and Their Second Fundamental Forms

Hiroshi Endo

Abstract We study the second fundamental form of invariant submanifolds of a contact metric manifold with ξ̄ belonging to the k-nullity distribution and with constant ϕ̄-sectional curvature. Then we shall show that some Kon’s results are special cases of ours.

Mathematics Subject Classification: 53C15 Key Words: contact metric manifold, compact minimal submanifold, Simons’ type formulae.

1 Introduction

In [12], Simons classified compact minimal submanifolds of a sphere into three categories with respect to the length of the second fundamental form. Then he got a very important formula (Simons’ type formula) with respect to minimal submanifolds (see [12], pp. 81). Kon [9] studied, by using Simons’ type formula, the pinching problem for the length of the second fundamental form of a compact invariant submanifold in Sasakian manifolds of constant ϕ̄-sectional curvatures and got the condition for its invariant submanifold to be totally geodesic. Moreover, he studied an invariant submanifold with trivial normal connection.

On the other hand, the present author [8] got the curvature tensor of a contact metric manifold with ξ̄ belonging to the k-nullity distribution and with constant ϕ̄-sectional curvature. The purpose of this paper is to lead Simons’ type formula for the second fundamantal form of invariant submanifolds of a contact metric manifold with ξ̄ belonging to the k-nullity distribution and with constant ϕ̄-sectional curvature and to show that some Kon’s results [9] are special cases of ours.

2 Contact metric manifolds

Let M̄ be an (2r + 1)-dimensional contact metric manifold and (ϕ̄, ξ̄, η̄, ḡ) be its contact metric structure. Then we have

Balkan Journal of Geometry and Its Applications, Vol.3, No.2, 1998, pp. 9-21 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

ϕ̄^{2} = −I + η̄ ⊗ ξ̄, ϕ̄η̄ = 0, η̄ ∘ ϕ̄ = 0, η̄(ξ̄) = 1,
< ϕ̄X̄, ϕ̄Ȳ >=< X̄, Ȳ > −η̄(X̄)η̄(Ȳ), < X̄, ξ̄ >= η̄(X̄),
dη̄(X̄, Ȳ) =< ϕ̄X̄, Ȳ >, X̄, Ȳ ∈ X (M̄),

where we denote the metric tensor field by <, > instead of ḡ and the Lie algebra

of vector fields on M̄ by X (M̄). On such a manifold we define h̄ = −\frac{1}{2}Lξ̄ϕ̄ (L

denotes the Lie differentiation). Then h̄ is symmetric, h̄ anti-commutes with ϕ̄ (i. e., ϕ̄h̄ + h̄ϕ̄ = 0), h̄ξ̄ = 0, η̄ ∘ h̄ = 0 and Tr h̄ = 0, where Tr h̄ is the trace of h̄. It is well-known that the vector field ξ̄ is a Killing vector field if and only if h̄ vanishes. Also

(2.1)  ¯∇X̄ ξ̄ = ϕ̄X̄ + ϕ̄h̄X̄ (and thus ¯∇ξ̄ξ̄ = 0),

where ¯∇ is the Riemannian connection of <, > (e.g., [7], cf. [1]). A contact metric manifold M̄ for wich ξ̄ is Killing is called a K-contact manifold. We also recall that on a K-contact manifold it is valid R̄(X̄, ξ̄)ξ̄ = X̄ − η̄(X̄)ξ̄, X̄ ∈ X (M̄), where R̄ is the curvature tensor of M̄. A contact structure on M̄^{2r+1} gives rise to an almost complex structure on the product M̄^{2r+1} × R, where R is the real line. If this almost complex structure is integrable, the contact metric manifold is said to be Sasakian. Equivalently, a contact metric manifold is Sasakian if and only if

R̄(X̄, Ȳ)ξ̄ = η̄(Ȳ)X̄ − η̄(X̄)Ȳ , X, Y ∈ X (M̄).

The k-nullity distribution (e.g., see [13]) of a Riemannian manifold (M, <, >) for a real number k is the distribution

N(k): p → N_{p}(k) = {Z̄ ∈ T_{p}(M̄)|R̄(X̄, Ȳ)Z̄ = k(< Ȳ , Z̄ > X̄− < X̄, Z̄ > Ȳ)}

for any X̄, Ȳ ∈ T_{p}(M̄). From now on, if we don’t refer something else, we suppose that M̄ is a contact metric manifold with ξ̄ belonging to the k-nullity distribution, i.e.,

(2.2)  R̄(X̄, Ȳ)ξ̄ = k(η̄(Ȳ)X̄ − η̄(X̄)Ȳ), X̄, Ȳ ∈ X (M̄).

In particular, if M̄ is Sasakian, then k = 1.

Then the following lemma is needed later ([11], [13] and [8]).

Lemma 2.1 Let M̄ be a contact metric manifold with ξ̄ belonging to the knullity distribution. Then we have

(2.3)  Q̄ξ̄ = (2nk)ξ̄, h̄^{2} = (k − 1)ϕ̄^{2} (and hence k ≤ 1);
(2.4) ( ¯∇X̄ ϕ̄)Ȳ = − < X̄ + h̄X̄, Ȳ > ξ̄+ η̄(Ȳ)(X̄ + h̄X̄) (and thus ¯∇ξ̄ϕ̄ = 0);

(2.5)

\begin{aligned}−( ¯∇X̄ h̄)Ȳ + ( ¯∇Ȳ h̄)X̄ & = & (1 − k)(2 < X̄, ϕ̄Ȳ > ξ̄+ η̄(X̄)ϕ̄Ȳ − η̄(Ȳ)ϕ̄X̄) \\ & + & η̄(X̄)ϕ̄h̄Ȳ − η̄(Ȳ)ϕ̄h̄X̄, \end{aligned}

for any vector fields X̄, Ȳ ∈ X (M̄), where Q̄ is the Ricci operator on M̄.

If X̄ is a unit vector which is orthogonal to ξ̄, we say that X̄ and ϕ̄X̄ span a ϕ̄-section. If the sectional curvature H̄(X̄) of all ϕ̄-sections is independent of X̄, we say that M̄ is of pointwise constant ϕ̄-sectional curvature.

Then we get the following theorem [8].

Theorem 2.1 Let M̄ be a contact metric manifold with ξ̄ belonging to the knullity distribution. If M̄ is of pointwise constant ϕ̄-sectional curvature H̄, then the curvature tensor has the following form: (2.6)

4R̄(X̄, Ȳ)Z̄ = (H̄ + 3)(< Ȳ , Z̄ > X̄− < X̄, Z̄ > Ȳ) + (H̄ − 1)(η̄(X̄)η̄(Z̄)Ȳ
− η̄(Ȳ)η̄(Z̄)X̄ + η̄(Ȳ) < X̄, Z̄ > ξ̄− η̄(X̄) < Ȳ , Z̄ > ξ̄+
+ < ϕ̄Ȳ , Z̄ > ϕ̄X̄+
− < ϕ̄X̄, Z̄ > ϕ̄Ȳ − 2 < ϕ̄X̄, Ȳ > ϕ̄Z̄ + 4(K − 1)(η̄(Ȳ)η̄(Z̄)X̄
− η̄(X̄)η̄(Z̄)Ȳ + η̄(X̄) < Ȳ , Z̄)ξ̄− η̄(Ȳ) < X̄, Z̄)ξ̄)
+ 4(< h̄Ȳ , Z̄ > X̄− < h̄X̄, Z̄ > Ȳ + < Ȳ , Z̄ > h̄X̄− < X̄, Z̄ > h̄Ȳ
+ η̄(X̄)η̄(Z̄)h̄Ȳ − η̄(Ȳ)η̄(Z̄)h̄X̄ + η̄(Ȳ) < h̄X̄, Z̄ > ξ̄
− η̄(X̄) < h̄Ȳ , Z̄ > ξ̄) + 2(< h̄Ȳ , Z̄ > h̄X̄− < h̄X̄, Z̄ > h̄Ȳ
+ < ϕ̄h̄X̄, Z̄ > ϕ̄h̄Ȳ − < ϕ̄h̄Ȳ , Z̄ > ϕ̄h̄X̄),

where H̄ is constant on M̄ if n ≠ 1.

3 Invariant submanifolds

Let M̄ = M̄^{2r+1}(ϕ̄, ξ̄, η̄, <, >) be a contact metric manifold. A submanifold M = M^{2n+1} of M̄ is said to be invariant if

(i) ξ̄ is tangent to M everywhere on M;

(ii) ϕ̄X is tangent to M for any tangent vector X to M.

It is well-known that any invariant submanifold M with induced structure tensor (ϕ, ξ, η, <, >) of M̄ is also contact metric manifold and is minimal in M̄ (e.g., [3], [4]). If we define an operator h = −\frac{1}{2}L_{ξ}ϕ in an invariant submanifold M(ϕ, ξ, η, <, >) of a contact metric manifold M̄(ϕ̄, ξ̄, η̄, <, >), then we have the results that h is symmetric, h anti-comutes with ϕ (i.e., ϕh + hϕ = 0), hξ = 0, η ∘ h = 0 and Tr h = 0. Moreover, by the definition of h̄, we can see that h̄X is tangent to M and h̄X = hX for X ∈ X (M) (see [5]).

Let X (M)^{⊥} be the set of all vector fields normal to M. We denote by ¯∇ the covariant differentiation in M̄ and ∇ the one in M determined by the induced metric on M. If we denote by A the second fundamental form of M, then the Gauss-Weingarten formulas are given by

¯∇_{X}Y = ∇_{X}Y + B(X, Y), ¯∇_{X}N = −A^{N}(X) + D_{X}N,

X, Y ∈ X (M), N ∈ X (M), where < A^{N}(X), Y >=< B(X, Y), N > and D is the linear connection in the normal bundle T (M)^{⊥}. The covariant derivative of B is given by

(∇_{X}B)(Y, Z) = D_{X}(B(Y, Z)) − B(∇_{X}Y, Z) − B(Y, ∇_{X}Z).

For X, Y ∈ X (M), the Gauss-Weingarten formulas implies

(3.1)  (R̄(X, Y)Z)^{T} = R(X, Y)Z − A^{B(Y,Z)}(X) + A^{B(X,Z)}(Y),

where T is the tangential projection on M. For any tangent vector field W on M we obtain the Gauss equation (3.2)

\begin{aligned}< (R̄(X, Y)Z, W > & = & < R(X, Y)Z, W > − < B(Y, Z), B(X, W) > + \\ & + & < B(X, Z), B(Y, W) > . \end{aligned}

On the other hand, using the Gauss-Weingarten formulas, we get

(3.3)  (R(X, Y)N)^{⊥} = R^{⊥}(X, Y) − B(A^{N}(Y), X) + B(A^{N}(X), Y),

where X, Y ∈ X (M), N ∈ X (M)^{⊥} and R^{⊥}(X, Y) = [D_{X}, D_{Y}] − D_{[X,Y]}. We need the following lemma later.

Lemma 3.1 [([3], [5])]. Let M be an invariant submanifold of a contact metric manifold M̄. Then the second fundamental form A of M satisfies A^{N}ϕ = −ϕA^{N}, A^{N}ξ = 0, N ∈ X (M)^{⊥}.

Let M̄ be a contact metric manifold with ξ̄ = ξ belonging to the k-nullity distribution. Then, by (2.2), we have

R̄(X, Y)ξ = k(η̄(Y)X − η̄(X)Y) = k(η(Y) − η(X)Y).

Here, from the Gauss equation and Lemma 3.1 we find R̄(X, Y)ξ = R(X, Y)ξ. Therefore we get R(X, Y)ξ = k(η(Y)X − η(X)Y), so that, M is also a contact metric manifold with ξ belonging to the k-nullity distribution. So, we have the following identities:

(3.4)  Qξ = 2nkξ, h^{2} = (k − 1)ϕ^{2} (and hence k ≤ 1);

(3.5)  (∇_{X}ϕ)Y = − < X + hX, Y > ξ + η(Y)(X + hX);

(3.6)

\begin{aligned}−(∇_{X}h)Y + (∇_{Y} h)X & = & (1 − k)(2 < X, ϕY > ξ + η(X)ϕY − η(Y)ϕX \\ & + & η(X)ϕhY − η(Y)ϕhX, \end{aligned}

for any vector fields X, Y ∈ X (M), where Q is the Ricci operator on M. Here we have the following.

Lemma 3.2 Let M̄ be a contact metric manifold with ξ̄ belonging to the knullity distribution. If M is an invariant submanifold of M̄, then the following are satisfied

(3.7)  ϕA^{N} = A^{ϕN};

(3.8)  ( ¯∇_{X}h)N = 0;

(3.9)  hA^{N} = A^{h̄N};

(3.10)  A^{N}h = hA^{N},  X, Y ∈ X (M), N ∈ X (M)^{⊥}.

Proof. At first we prove (3.7). From the Gauss formula we see that

(3.11)  ϕ̄( ¯∇_{X}Y) = ϕ(∇_{X}Y) + ϕ̄B(X, Y).

On the orher hand, by (2.4) and the Gauss formula, we find

(3.12)  \begin{aligned}ϕ̄( ¯∇_{X}Y) & = & ¯∇_{X}(ϕ̄Y) − ( ¯∇_{X} ϕ̄)Y \\ & = & ¯∇_{X}(ϕY)+ < X + hX, Y > ξ − η(Y)X − η(Y)hX \\ & = & (∇_{X}ϕ)Y + ϕ(∇_{X}Y) + B(X, ϕY) \\ & + & < X + hX, Y > ξ − η(Y)X − η(Y)hX. \end{aligned}

Taking the normal parts of (3.11) and (3.12), we get ϕ̄B(X, Y) = B(X, ϕY). Thus we find ϕA^{N} = A^{ϕN}.

Next we prove (3.8) and (3.9). By Weingarten formula, we get the following two equations:

(3.13)  ¯∇_{X}(h̄N) = −A^{h̄N}(X) + D_{X}(h̄N);

(3.14)  \begin{aligned}¯∇_{X}(h̄N) & = & ( ¯∇_{X} h̄)N + h̄( ¯∇_{X}N) \\ & = & ( ¯∇_{X} h̄)N + h̄(−A^{N}(X) + D_{X}N) \\ & = & ( ¯∇_{X} h̄)N − h̄A^{N}(X) + h̄D_{X}N). \end{aligned}

On the other hand, we have

L_{ξ}( ¯∇_{X} ϕ̄)N − ¯∇_{X}(L_{ξ}ϕ̄)N − ( ¯∇_{[ξ,X]}ϕ̄)N = 0,

from which, we obtain

( ¯∇_{X} h̄)N = −\frac{1}{2}L_{ξ}( ¯∇_{X} ϕ̄)N + \frac{1}{2}( ¯∇_{[ξ,X]}ϕ̄)N.

Here, from (2.4), we see that ( ¯∇_{X} ϕ̄)N = ( ¯∇_{[ξ,X]}ϕ̄)N = 0. Thus we get ( ¯∇_{X} h̄)N = 0. Taking the tangential parts of (3.13) and (3.14), we find our result.

Last we prove (3.10). Using the Gauss formula, we get

(3.15)  \begin{aligned}¯∇_{X}(h̄Y) & = & ( ¯∇_{X} h̄)Y + h̄( ¯∇_{X}Y) \\ & = & ( ¯∇_{X} h̄)Y + h̄(∇_{X}Y + B(X, Y)) \\ & = & ( ¯∇_{X} h̄)Y + h̄(∇_{X}Y) + h̄(B(X, Y)). \end{aligned}

On the other hand, we find

(3.16)  \begin{aligned}¯∇_{X}(h̄Y) & = & ¯∇_{X}(hY) \\ & = & ∇_{X}(hY) + B(X, hY) \\ & = & (∇_{X}h)Y + h(∇_{X}Y) + B(X, hY). \end{aligned}

From (3.15) and (3.16), we have

(3.17)  ( ¯∇_{X} h̄)Y + h̄(B(X, Y)) = (∇_{X}h)Y + B(X, hY).

Interchanging X and Y in the above equation, we find

(3.18)  ( ¯∇_{Y} h̄)X + h̄(B(Y, X)) = (∇_{Y} h)X + B(Y, hX).

Subtracting (3.17) from (3.18) and using (2.5), we obtain B(X, hY) = B(Y, hX), from which A^{N}h = hA^{N}.

Using Lemma 3.2, we get the following lemma.

Lemma 3.3 Let M̄^{2r+1} be a contact metric manifold with ξ̄ belonging to the knullity distribution. If M^{2n+1} is invariant submanifold of M̄^{2r+1}. Then we have Tr hA^{2} = 0 and (Tr ha)^{2} ≤ 2n(1 − k)Tr A^{2}.

Proof. At first, h is represented by the following matrix form (see [13], pp. 446):

h = ( \begin{matrix}μ_{1} & & & & & & & & & & 0 \\ & . & . & . \\ & & & & μ_{n} \\ & & & & & −μ_{1} \\ & & & & & & . & . & . \\ & & & & & & & & & −μ_{n} \\ 0 & & & & & & & & & & 0 \end{matrix} )

where μ_{i} = \sqrt{1 − k}, (1 ≤ i ≤ n). From (3.10) we can take the same orthogonal matrix as h’s one to orthogonalize A. Therefore, from Lemma 3.1, A is expressed as in the following:

A = ( \begin{matrix}ν_{1} & & & & & & & & & & 0 \\ & . & . & . \\ & & & & ν_{n} \\ & & & & & −ν_{1} \\ & & & & & & . & . & . \\ & & & & & & & & & −ν_{n} \\ 0 & & & & & & & & & & 0 \end{matrix} )

Thus, we get Tr hA = 2\sqrt{1 − k}(ν_{1} + ⋯ + ν_{n}) and Tr hA^{2} = 0, from which

(Tr hA)^{2} = 4(1 − k)(ν_{1} + ⋯ + ν_{n})^{2}
≤ 4n(1 − k)(ν_{1}^{2} + ⋯ + ν_{n}^{2})
= 2n(1 − k)Tr A^{2}.

The following two theorems are known ([5], [2]).

Theorem 3.1 Let M be an invariant submanifold of a contact manifold M̄. Then A^{N} = A^{N}h if and only if (∇_{X}A^{N})ξ = 0.

Theorem 3.2 Let M^{2n+1} be a contact metric manifold with R(X, Y)ξ = 0 for all vectors X and Y. Then M^{2n+1} is locally the Riemannian product of a flat (n + 1)-dimensional manifold and an n-dimensional manifold of positive curvature 4.

By Theorem 3.2, we get the following lemma.

Lemma 3.4 Let M̄^{2r+1} be a contact metric manifold with ξ̄ belonging to the k-nullity distribution, and let M^{2n+1} be an invariant submanifold of M̄. If the second fundamental form A of M^{2n+1} is covariant constant, them either M^{2n+1} is totally geodesic, or M^{2n+1} is locally the Riemannian product of a flat (n + 1)- dimensional manifold and an n-dimensional manifold of constant curvature 4.

Proof. By Theorem 3.1 and out assumption, we get A^{N} = A^{N}h. Therefore, from Lemma 2.1, i.e. follows that A^{n} = A^{N}h = A^{N}h^{2} = A^{N}(k − 1)ϕ^{2}. By Lemma 3.1 we find

A^{N}X = (k − 1)A^{N}(−X + η(X)ξ) = (1 − k)A^{N}X.

This implies kA^{N}X = 0, from which k = 0 or A^{N}X = 0. In the case of A^{N}X = 0, X ∈ X (M), M^{2n+1} is totally geodesic. When we have k = 0, by Theorem 3.2, M^{2n+1} is locally the Riemannian product of a flat (n + 1)-dimensional manifold and an n-dimensional manifold of constant curvature 4.

We now suppose that the ambient manifold M̄ is a contact metric manifold with ξ̄ belonging to the k-nullity distribution and with constant ϕ̄-sectional curvature H̄. Then we have (3.19)

R(X, Y)Z = \frac{H̄ + 3}{4}(< Y, Z > X− < X, Z > Y) + \frac{H̄ − 1}{4}(η(X)η(Z)Y
−η(Y)η(Z)X + η(Y) < X, Z > ξ − η(X) < Y, Z > ξ
+ < ϕY, Z > ϕX− < ϕX, Z > ϕY − 2 < ϕX, Y > ϕZ)
+(k − 1)(η(Y)η(Z)X − η(X)η(Z)Y + η(X) < Y, Z > ξ
−η(Y) < X, Z > ξ)+ < hY, Z > X− < hX, Z > Y
+ < Y, Z > hX− < X, Z > hY + η(X)η(Z)hY
−η(Y)η(Z)hX + η(Y) < hX, Z > ξ − η(X) < hY, Z > ξ)
+\frac{1}{2}(< hY, Z > hX− < hX, Z > hY + < ϕhX, Z > ϕhY
− < ϕhY, Z > ϕhX) + A^{B(Y,Z)}X − A^{B(X,Z)}Y,

(3.20)  (∇_{X}B)(Y, Z) − (∇_{Y} B)(X, Z) = 0,

(3.21)  \begin{aligned}Ric(Y, Z) & = & (\frac{n(H̄ + 3) + (H̄ − 1)}{2} + (k − 1)) < Y, Z > + \\ & + & ((2n − 1)(k − 1) − \frac{(n + 1)(H̄ − 1)}{2}) η(Y)η(Z)+ \\ & + & 2(n − 1) < hY, Z > − ∑_{i} < B(X, e_{i}), B(Y, e_{i}) >, \end{aligned}

(3.22) S_{c} = n^{2}(H̄ + 3) + n(H̄ + 1) + 4n(k − 1) − ∑_{i,j} < B(e_{i}, e_{j}), B(e_{i}, e_{j}) >,

where {e_{i}} is an orthonormal basis of M, Ric is the Ricci curvature on M and S_{c} is the scalar curvature on M (see [8]).

4 Simon’s type formula of invariant submanifolds

In this section, we assume that M̄ = M̄^{2r+1}(ϕ̄, ξ̄, η̄, <, >) be a contact metric manifold with ξ̄ belonging to the k-nullity distribution and with pointwise constant sectional curvature H̄, and let M = M^{2n+1}(ϕ, ξ, η, <, >) be an invariant submanifold of M̄. We put p = r−n. First we have (2.6). Since any invariant submanifold M of a contact metric manifold M̄ is minimal, its second fundamental form A satisfies

(4.1)  ∇^{2}A = −A ∘ Ã − A_{∗} ∘ A + R̄(A) + R̄′,

where R̄(A) and R̄′ are defined by setting (4.2)

< R̄(A)^{W} (X), Y > = ∑_{i=1}^{2n+1}{2 < R̄(e_{i}, Y)B(X, e_{i}), W > +
+ 2 < R̄(e_{i}, X)B(Y, e_{i}), W > −
− < A^{W} (X), R̄(e_{i}, Y)e_{i} > − < A^{W} (Y), R̄(e_{i}, X)e_{i} >
< R̄(e_{i}, B(X, Y))e_{i}, W > −2 < A^{W} (e_{i}), R̄(e_{i}, X)Y >},

< R̄′(X), Y >= ∑_{i=1}^{2n+1} (< ¯∇_{X}(R̄)(e_{i}, Y)e_{i}, W > + < ¯∇_{e_{i}}(R̄)(e_{i}, X)Y, W >) ,

where X, Y ∈ T_{m}(M), W ∈ T_{m}(M)^{⊥} and e_{1}, …, e_{2n+1} is a frame in T_{m}(M) (see [12], pp. 81). The operator Ã is defined by

(4.3)  Ã = A^{t} ∘ A,

where A^{t} is the transpose of A. Let v_{1}, v_{2}, …, v_{2p} be a frame for T_{m}(M)^{⊥}. Then the operator A_{∗} is defined by setting A_{∗} = ∑_{i=1}^{2p} ad A^{i}ad A^{i}. Here we denote a^{v_{i}} by A^{i} to simplify. Ã and A_{∗} are symmetric and positive semi-definite operators.

Since M̄ is of constant ϕ̄-sectional curvature H̄, from (2.1), (2.6), (3.3), (3.4), Lemma 3.1 and Lemma 3.2 we get, after lengthly computation,

<R̄(A^{W})(X), Y >+<R̄^{′W} (X), Y > = \frac{(n+2)H̄+3n−2(k−1)}{2} < A^{W} X, Y >
+ 2n < hA^{W} X, Y >−2TrhA^{W} <X, Y >
+ 2η(X)η(Y)TrhA^{W} − TrhA^{W} <hX, Y > .

Hence we get by (4.1) and Lemma 3.1 (4.4)

\begin{aligned}− < ∇^{2}A, A > & = & < A ∘ Ã, A > + < A_{∗} ∘ A, A > −\frac{(n+2)H̄+3n−2(k−1)}{2}∥A∥^{2} \\ & ; & −2n ∑_{W} Tr h(A^{W})^{2} + ∑_{W}(Tr hA^{W})^{2}, \end{aligned}

where ∥A∥ denotes the length of the second fundamental from A. From (4.3) Lemma 3.2 we can see easily that ϕ̄Ã = Ãϕ̄. And Ã is symmetric, positive semi-definite at each point m ∈ M. Hence we can choose an orthogonal basis v_{1}, …, v_{p}, ϕ̄v_{1}, …, ϕ̄v_{p} of T_{m}(M)^{⊥} with respect to which the matrix form of Ã is of the form

A = ( \begin{matrix}a_{1}^{2} & & & & & & & & & 0 \\ & . & . & . \\ & & & & a_{p}^{2} \\ & & & & & a_{1}^{2} \\ & & & & & & . & . & . \\ 0 & & & & & & & & & a_{p}^{2} \end{matrix} )

We can see that

\frac{1}{2p}(2 ∑_{t=1}^{p} a_{t}^{2})^{2} ≤< A ∘ Ã, A >= 2 ∑_{t=1}^{p} a_{t}^{4} = 2((∑_{t=1}^{p} a_{t}^{2})^{2} − ∑_{t≠s}^{p} a_{t}^{2}a_{s}^{2})

(also see [12], pp. 93, pp. 94). From this we have

(4.5)  \frac{1}{2p}∥A∥^{4} ≤< A ∘ Ã, A >≤ \frac{1}{2}∥A∥^{4}.

Here we define the operator A^{∗} = ∑_{i=1}^{2p}(A_{i})^{2} which is also a symmetric, positive

semi-definite operator. Taking a basis e_{1}, …, e_{n}, ϕe_{1}, …, ϕe_{n}, ξ ∈ T_{m}(M), by Lemma 3.1 and (3.7) we have ϕA^{a}A^{b}ϕA^{a} = −A^{a}A^{b}A^{a}. Therefore we obtain

< A_{∗} ∘ A, A >= ∑_{a,b=1}^{2p} ∥[A^{a}, A^{b}]∥^{2} = 2Tr (A^{∗})^{2}.

Since A^{∗} is symmetric positive semi-definite and ϕA^{∗} = A^{∗}ϕ, by using a suitable frame A^{∗} is represented by the matrix form

A^{∗} = ( \begin{matrix}λ_{1} & & & & & 0 \\ & . & . & . \\ & & & & λ_{2n} \\ 0 & & & & & 0 \\ , \end{matrix} )

where λ_{n+t} = λ_{t}, λ_{t} ≥ 0. Then we have

< A_{∗} ∘ A, A >= 2 ∑_{i=1}^{2n} λ_{i}^{2} ≥ \frac{1}{n}(∑_{i=1}^{2n} λ_{i})^{2} = \frac{1}{n}∥A∥^{4},

\begin{aligned}< A_{∗} ∘ A, A > & = & 2((∑_{i=1}^{2n} λ_{i})^{2} − ∑_{i≠j}^{2n} λ_{i}λ_{j}) \\ & = & 2(∑_{i=1}^{2n} λ_{i})^{2} − 8 ∑_{t≠s}^{n} λ_{t}λ_{s}− < A_{∗} ∘ A, A > . \end{aligned}

Consequently we get the following inequality

(4.6)  \frac{1}{n}∥A∥^{4} ≤< A_{∗} ∘ A, A >≤ ∥A∥^{4}.

Therefore (4.4), (4.5), and (4.6) and Lemma 3.3 imply

(4.7) − < ∇^{2}A, A >≤ [\frac{3}{2}∥A∥^{2} − \frac{1}{2}((n + 2H̄ + 3n) + (2n − 1)(1 − k)]∥A∥^{2}.

If M is compact, then we get

− ∫_{M} < ∇^{2}A, A >= ∫_{M} < ∇A, ∇A > .

Theorem 4.1 Let M̄^{2r+1} be a contact metric manifold with ξ̄ belonging to the k-nullity distribution and with constant ϕ̄-sectional curvature H̄, and let M^{2n+1} be a compact invariant submanifold of M̄^{2r+1}. Then either M^{2n+1} is totally geodesic in M̄^{2r+1}, or M^{2n+1} is locally the Riemannian product of a flat (n+1)- dimensional manifold and an n-dimensional manifold of constant curvature 4, or at some point m ∈ M^{2n+1} the following inequality holds

∥A∥^{2}(m) > \frac{1}{3}((n + 2)H̄ + 3n − 2(2n − 1)(1 − k)).

Proof. From (4.7), we get

∫_{M}[\frac{3}{2}∥A∥^{2} − \frac{1}{2}((n + 2)H̄ + 3n) + (2n − 1)(1 − k)]∥A∥^{2} ≥ ∫_{M} < ∇A, ∇A >≥ 0.

Suppose ∥A∥^{2} ≤ \frac{1}{3}((n + 2)H̄ + 3n − 2(2n − 1)(1 − k)) everywhere. Then the

second fundamental form of M^{2n+1} is covariant constant, from which M^{2n+1} is totally geodesic, or M^{2n+1} is locally the Riemannian product of a flat (n + 1)- dimensional manifold and an n-dinemsional manifold of constant curvature 4. Therefore, except for this possibility, at some point m ∈ M^{2n+1},

∥A∥^{2}(m) > \frac{1}{3}((n + 2)H̄ + 3n − 2(2n − 1)(1 − k)).

If M̄^{2r+1} is of constant ϕ̄-sectional curvature H̄, then the scalar curvature S_{c} of M^{2n+1} is given by (3.22). Consequently we get the following corollary.

Corollary 4.1 Under the same assumption as in T heorem4.1, M^{2n+1} is totally geodesic in M̄^{2r+1}, or M^{2n+1} is locally the Riemannian product of a flat (n+1)- dimensional manifold and an n-dimensional manifold of constant curvature 4, or at some point m ∈ M,

S_{c}(m) < n^{2}(H̄ +3)+n(H̄ +1)+4n(k −1)− \frac{1}{3}((n+2)H̄ +3n−2(2n−1)(1−k)).

Remark 4.1 In the case of which M̄^{2r+1} is Sasakian, we have k = 1. Then Theorem 4.1 and Corollary 4.1 become the results of Kon ([9], pp.136).

5 Invariant submanifolds with trivial normal connection

In this section, we study, in the same way as Kon leads, invariant submanifolds with trivial normal connection. Let M̄ be a contact metric manifold with ξ̄ belonging to the k-nullity distribution and with constant ϕ̄-sectional curvature H̄, and let M be an invariant submanifold of M̄. Throughout in this section, we assume that the normal connection of M trivial, i. e., R^{⊥} = 0. By the assumption and (3.3) we obtain

< (R̄(X, ϕY)V)^{⊥}, ϕ̄V >= − < B(A^{V} (ϕY), X), ϕV > + < B(A^{V} (X), ϕY), ϕ̄V >,

where X, Y ∈ T_{m}(M) and V is a unit vector in T_{m}(M)^{⊥}. From (2.6) the left hand side of this becomes

< (R̄(X, ϕY)V)^{⊥}, ϕ̄V >= \frac{1}{2}(1 − H̄) < ϕX, ϕY >,

and by Lemma 3.1 and Lemma 3.2 we see that

− < B(A^{V} (ϕY), X)ϕ̄V > + < B(A^{V} (X), ϕY), ϕ̄V >= 2 < A^{V} (X), A^{V} (Y) > .

Hence we get

(5.1)  (1 − H̄) < ϕX, ϕY >= 4 < A^{V} (X), A^{V} (Y) > .

From this we obtain the following result.

Proposition 5.1 Let M̄ be a contact metric manifold with ξ̄ belonging to the k-nullity distribution and with constant ϕ̄-sectional curvature H̄. If the normal connection of an invariant submanifold M of M̄ is trivial, then H̄ ≤ 1 and equality holding if and only if M is totally geodesic in M̄.

Lemma 5.1 Let M̄ be a contact metric manifold with ξ̄ belonging to the knullity distribution and with constant ϕ̄-sectional curvature H̄, and let M be an invariant submanifold of M̄ with trivial normal connection. Let V and W be unit vectors in T_{m}(M)^{⊥}. If V is orthogonal to W, then

(5.2)  A^{V} A^{W} + A^{W} A^{V} = 0.

Proof. We may construct a new unit vector (V + W)/\sqrt{2} in T_{m}(M)^{⊥}. In view of (5.1) we see that for any vectors X, Y ∈ T_{m}(M)

< A^{V} A^{W} (X), Y > + < A^{W} A^{V} (X), Y >= 0,

which proves (5.2).

Theorem 5.1 Let M̄^{2r+1} be a contact metric manifold with ξ̄ belonging to the k-nullity distribution and with constant ϕ̄-sectional curvature H̄. If M^{2n+1} is invariant submanifold of M̄^{2r+1} and the codimension of M is greater than 2, then the following conditions are equivalent:

(I) the normal connection of M is trivial, i.e., R^{⊥} = 0; (II) H̄ = 1 and M is totally geodesic in M̄.

Proof. From (2.6) and (3.3) it is clear that the condition (II) implies the condition (I). Let us assume the condition (I) and that M is not totally geodesic in M̄. Let e_{1}, …, e_{2n}, ξ be a ϕ-basis in T_{m}(M) such that e_{n+i} = ϕe_{i}. If A^{V} (e_{i}) = 0 for some unit V ∈ T_{m}(M), then (5.1) implies that M is totally geodesic. Therefore A^{V} (e_{i}) ≠ 0 for any V and e_{i}. From (5.1) we can see that A(e_{1}), …, A(e_{2n} are linearly independent. On the other hand, by (3.3) and (5.2) we get

< (R̄(X, Y)V)^{⊥}, W >= 2 < A^{V} (X), A^{W} (Y) >,

where X, Y ∈ T_{m}(M) and W is a unit vector in T_{m}(M)^{⊥} which is orthogonal to V. Using (2.6), we obtain

< (R̄(X, Y)V)^{⊥}, W >= \frac{1}{2} << X, ϕY > ϕ̄V, W > .

Hence we have

(5.3)  (1 − H̄) << X, ϕY > ϕ̄V, W) = 4 < A^{V} (X), A^{W} (Y) > .

If 2p > 2, we can take W which is orthogonal to V and ϕ̄V. Then regarding to (5.3), it follows that < A^{V} (X), A^{V} (Y) >= 0 for any X, Y ∈ T_{m}(M). And by the assumption, A^{W} (e_{i}) ≠ 0 for any i (i = 1, …, 2n). Therefore A^{V} (e_{i}) is orthogonal to A^{W} (e_{j}) for any i, j. Consequently A^{V} (e_{1}), …, A^{W} (e_{2n}), A^{W} (e_{1}), …, A^{W} (e_{2n}) are linearly independent. But each A^{V} (e_{i}) and A^{W} (e_{i}) are in T_{m}(M) and we have dim T_{m}(M) = 2n + 1. This is the contradiction. Therefore M is totally geodesic in M̄ and hence H̄ = 1.

Remark 5.1 In the case of which M̄ is Sasakian, Theorem 5.1 turns out a result of Kon ([9], pp.138).

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