Submanifolds in Normal Complex Contact Manifolds

In the present article we initiate the study of submanifolds in normal complex contact metric manifolds. We define invariant and anti-invariant ( C C -totally real) submanifolds in such manifolds and start the study of their basic properties. Also, we establish the Chen first inequality and Chen inequality for the invariant δ ( 2 , 2 ) for C C -totally real submanifolds in a normal complex contact space form and characterize the equality cases. We also prove the minimality of C C -totally real submanifolds of maximum dimension satisfying the equalities.


Introduction
Complex and contact geometries represent some of the most studied areas in differential geometry. The complex contact metric structures are less explored and there is a short list of papers in the mathematical literature on this topic. However, in [1] there is a chapter in which D.E. Blair realizes a comprehensive survey of known results on (normal) complex contact manifolds.
In [2], D.E. Blair and the first author in this work, proved that a locally symmetric normal complex contact metric manifold is locally isometric to the complex projective space CP 2n+1 (4) of constant holomorphic curvature 4. They also studied reflections in the integral submanifolds of the vertical subbundle of a normal complex contact manifold and showed that when such reflections are isometries the manifold fibers over a locally symmetric space. If the normal complex contact manifold is Kähler, then the manifold fibers over a quaternionic symmetric space. Also, if the complex contact structure is given by a global holomorphic contact form, then the manifold fibers over a locally symmetric complex symplectic manifold.
In [3], the same authors studied the homogeneity and local symmetry of complex (κ, µ)-spaces. It was proved that for k < 1, a complex (κ, µ)-space is locally homogeneous and GH-locally symmetric.
On the other hand, in differential geometry, the theory of submanifolds plays a very important role. As we know, articles on submanifolds in normal complex contact manifolds have not been published until now.
In this paper, we define invariant and anti-invariant (CC-totally real) submanifolds of normal complex contact metric manifolds and start the study of their basic properties.
Also, the theory of Chen invariants represents one of the most useful tools for obtaining relationships between extrinsic and intrinsic invariants of a submanifold. In [4], B.-Y. Chen wrote a survey on Chen invariants and Chen inequalities and their applications. We establish the Chen first inequality and Chen inequality for the invariant δ (2, 2) for CC-totally real submanifolds in a normal complex contact space form and give the characterizations of the equality cases.

Preliminaries
A complex contact manifold [1] is a complex manifold M of odd complex dimension 2n + 1 together with an open covering {O} of coordinate neighborhoods such that: The complex contact structure determines a non-integrable subbundle H by the equation θ = 0; H is called the complex contact subbundle or the horizontal subbundle.
A Hermitian manifold M with almost complex structure J, Hermitian metric g, and open covering by coordinate neighborhoods {O} is a complex almost contact metric manifold if it satisfies the following two conditions: (1) In each O there exist 1-forms u and v = u • J with dual vector fields U and V = −JU and (1,1) tensor fields G and H = GJ such that where A and B are functions with A 2 + B 2 = 1.
A complex contact manifold admits a complex almost contact metric structure for which the local contact form θ is u − iv to within a non-vanishing complex-valued function multiple. The local tensor fields G and H and du and dv are related by for some 1-form σ and where G(X, Y) = g(X, GY) and H(X, Y) = g(X, HY).
Also, on O ∩ O one has U ∧ V = U ∧ V; it follows that there is a global vertical bundle V orthogonal to H (which is generally assumed to be integrable), and, in this case, σ takes the form σ(X) = g(∇ X U, V), ∇ being the Levi-Civita connection of g. The subbundle V is the analogue of the characteristic or Reeb vector field of real contact geometry.
A complex contact manifold with a complex almost contact metric structure satisfying these conditions is called a complex contact metric manifold. (iv) C n+1 × CP n (16).
S. Ishihara and M. Konishi [5] introduced a notion of normality for complex contact structures. They defined two tensor fields S and T by where [G, G] and [H, H] denote the Nijenhuis tensors of G and H respectively.
A complex contact structure is said to be normal if S and T vanish, S = T = 0. This notion is too strong; among its implications is that the underlying Hermitian manifold (M, g) is Kähler. Also one remarks that the canonical examples of a complex contact manifold, the odd-dimensional complex projective space, is normal in this sense, however, the complex Heisenberg group is not. B. Korkmaz [6] generalized the notion of normality and we will use her definition here. With this notion of normality both odd-dimensional complex projective space and the complex Heisenberg group with their standard complex contact metric structures are normal.
A complex contact metric structure is normal [6] if The definition appears to depend on the special nature of U and V, but it respects the change in overlaps, O ∩ O ; then it is a global notion.
The expressions for the covariant derivatives of the structures tensors on a normal complex contact metric manifold are Equivalently, a complex contact metric manifold is normal if and only if the covariant derivatives of G and H have the following forms: For the Hermitian structure J we have B. Korkmaz (see [1,6]) defined the notion of GH-sectional curvature for a (normal) complex contact metric manifoldM.
For p ∈M and a unit vector X ∈ H p , the plane in T pM spanned by X and Y = λ GX + µHX, λ, µ ∈ R, λ 2 + µ 2 = 1, is called a GH-plane section. The sectional curvature of the GH-plane section is denoted byK(X, Y).
For a given tangent vector X, the sectional curvatureK(X, Y) is independent of the vector Y in the plane of GX and HX. This is equivalent toK(X, GX) =K(X, HX) andg(R X(GX) HX, X) = 0.
LetM be a normal complex contact manifold. If the GH-sectional curvature is independent of the choice of GH-section at each point, it is constant on the manifoldM and thenM is called a (normal) complex contact space form.
According to [6] and the convention used by D.E. Blair, the curvature tensor ofM have the expressionR where Ω (U, V) = g(U, JV).

Remark 1 ([1]).
(i) An odd-dimensional complex projective space with the Fubini-Study metric of constant holomorphic sectional curvature 4 is of constant GH-sectional curvature 1. (ii) The complex Heisenberg group has holomorphic sectional curvature 0 for horizontal and vertical holomorphic sections and constant GH-sectional curvature −3.
We recall some results.
LetM be a normal complex contact metric manifold. ThenM has constant GH-sectional curvature c if and only if for X horizontal, the holomorphic sectional curvature of the plane spanned by X and JX is c + 3.
LetM be a normal complex contact metric manifold of constant GH-sectional curvature +1 and satisfying dσ (V, U) = 2. ThenM has constant holomorphic sectional curvature c.
If, in addition,M is complete and simply connected, thenM is isometric to CP 2n+1 with the Fubini-Study metric of constant holomorphic sectional curvature c.
We denote a (normal) complex contact space form byM(c), c being the constant GH-sectional curvature ofM.

Submanifolds
By analogy with the geometry of submanifolds in (real) contact manifolds [7], we shall define certain special classes of submanifolds in normal complex contact manifolds (see also [8]).
Let M be a submanifold of a normal complex contact manifoldM. Assume that U and V are normal vector fields to M. For p ∈ M and X, Y ∈ T p M, we have g(GX, Y) = −g(∇ X U, Y) = g(U,∇ X Y) = g(U, h(X, Y)).
Since the first term of the above equalities is skew-symmetric and the last term is symmetric (in X, Y), then g(GX, Y) = 0. Similarly, g(HX, Y) = 0.
Then we have the following. Based on the previous two remarks, we define two classes of submanifolds.  We shall prove the minimality of an invariant submanifold in a normal complex complex contact manifold.
First of all, we remark that U and V have to be tangent to the submanifold. Suppose that U is not tangent to the submanifold and decompose U into its tangential and normal parts, say U = U T + U ⊥ . Then 0 = GU = GU T + GU ⊥ ; since the G-invariance of the tangent space implies the G invariance of the normal space, both GU T and GU ⊥ vanish.
Using the Formula (3) for the covariant derivative of G for a normal complex contact metric manifold, this becomes Comparing with (7), g(U T , U T ) = 1, i.e., U T is unit as is U; therefore U ⊥ = 0, contradicting the supposition that U was not tangent.
For this reason, an orthonormal basis of M, dimM = 4n + 2, can be written as The proof of the minimality uses the formula of the covariant derivative given by B. Foreman in his Ph.D. Thesis [9].
Let pr : TM −→ H denote the projection to the horizontal subbundle and J = pr • J. We then have Now taking X as horizontal and Y = X we make the following computation

Example 2. Example of an invariant submanifold:
Consider the Segre embedding: Its image is an invariant submanifold.
Another important class of submanifolds are the anti-invariant submanifolds.

Definition 2.
A submanifold M of a normal complex contact manifoldM is said to be a CC-totally real (anti-invariant) submanifold if (i) U and V are normal to M; (ii) M is a totally real submanifold ofM (with respect to J).
For a CC-totally real (anti-invariant) submanifold M of a (normal) complex contact space form M(c) (dim R M = n; dim RM (c) = m) of arbitrary codimension, an orthonormal basis of T p M, p ∈ M, can be written as {E 1 , E 2 , . . . , E n }, n = dim R M. Obviously,

Chen Inequalities
In this section we shall prove the Chen first inequality and Chen inequality for the invariant δ (2, 2) for CC-totally real submanifolds in a normal complex contact space formM(c).
Let (M, g) be a Riemannian manifold of dimension n ≥ 2 and denote by K and τ the sectional curvature and scalar curvature of M, respectively.
where π 1 and π 2 are mutually orthogonal plane sections.
In order to prove Chen first inequality, we state the following lemma.

Lemma 1 ([11]
). Let n ≥ 3 be an integer and let a 1 , ..., a n be n real numbers. Then one has: Moreover, the equality holds if and only if a 1 + a 2 = a 3 = ... = a n .
Let M be an n-dimensional CC-totally real submanifold of a normal complex contact space form of arbitrary codimension m.
The Gauss equation for M inM(c) is given bỹ for any X, Y, Z, W ∈ Γ(TM), where h is the second fundamental form. Let p ∈ M, π a plane section in T p M, {E 1 , E 2 } an orthonormal basis of π, {E 1 , ..., E n } an orthonormal basis of T p M and {E n+1 , ..., E m } an orthonormal basis of T ⊥ p M. The Gauss equation implies .., n}, α ∈ {n + 1, ..., m}.
On the other hand, Subtracting the above two equations, one has By applying Lemma 1, we obtain for all α ∈ {n + 1, ..., m}: By summing the above relations, we obtain where H is the mean curvature vector.
The equality holds at a point p ∈ M if and only if for any α ∈ {n + 1, ..., m}, If we take E n+1 parallel to H(p) and E 1 , E 2 such that h n+1 12 = 0, the shape operators take the forms given in the following: Moreover, the equality case of the inequality holds at a point p ∈ M if and only if there exist an orthonormal basis {E 1 , E 2 , ..., E n } of T p M and an orthonormal basis {E n+1 , ..., E m } of T ⊥ p M such that the shape operators take the following forms: If M is of maximum dimension, i.e., dim M = n and dimM = 4n + 2 (the analogue of a Legendrian submanifold in the real case), we obtain: Theorem 5. LetM(c) be a normal complex contact space form of dimension 4n + 2. Then any n-dimensional CC-totally real submanifold M satisfying the equality case of Chen first inequality, identically, is minimal.
Proof. Let M be an n-dimensional CC-totally real submanifold in a (4n + 2)-dimensional normal complex contact space formM(c) satisfying the equality case of Chen first inequality, identically. Then the Equations (11) can be written as Then, by using (1), we have On the other hand, using (3), we find for any j = 3 It follows by (12) that i.e., M is a minimal submanifold.
Next we shall prove Chen inequality for the invariant δ (2, 2). We shall use the following Lemma.

Lemma 2 ([12]
). Let n ≥ 4 be an integer and let a 1 , ..., a n be n real numbers. Then one has: Moreover, the equality holds if and only if a 1 + a 2 = a 3 + a 4 = a 5 = ... = a n .
For a CC-totally real submanifold of maximum dimension, we have Theorem 7. LetM(c) be a normal complex contact space form of dimension 4n + 2. Then any n-dimensional CC-totally real submanifold M satisfying the equality case of Chen inequality for δ (2, 2), identically, is minimal.
Its proof is similar to the proof of Theorem 5.