C-Totally Real Submanifolds of R^{2n+1} Satisfying a Certain Inequality

Dragoş Cioroboiu

Abstract

We establish a sharp inequality between the squared mean curvature and the scalar curvature for a C-totally real submanifold of maximum dimension in a Sasakian space form. In particular we investigate C-totally real submanifolds of R^{2n+1} satisfying the equality case.

Mathematics Subject Classification: 53C12, 53C25 Key words: Sasakian space form, curvature, Whitney immersion, C-totally real submanifold

1 Introduction

Let C^{n} denote the complex Euclidean n−space with complex structure J defined by

J(x_{1}, x_{2}, ..., x_{2n}) = (−x_{n+1}, ..., −x_{2n}, x_{1}, ..., x_{n}).

If f : M ⟶ C^{n} is an isometric immersion from a Riemannian n−manifold M into C^{n}, then M is called a Lagrangian submanifold (or totally real submanifold in [5] ) if J carries each tangent space of M into its normal space. Lagrangian submanifolds appear naturally in the context of classical mechanics and mathematical physics.

It is well-known, that every curve in C^{1} is Lagrangian. For n ≥ 2, there is a Lagrangian immersion from an n−sphere S^{n} into C^{n} given by Whitney which is a called the Whitney immersion. The Whitney immersion is defined as follows :

Let f : E^{n+1} ⟶ C^{n} be a map from E^{n+1} into the complex Euclidean space C^{n} defined by :

f(x_{0}, x_{1}, ..., x_{n}) = \frac{1}{1 + x_{0}^{2}} (x_{1}, ..., x_{n}, x_{0}x_{1}, ..., x_{0}x_{n}).

Denote by S^{n} the unit hypersphere of E^{n+1} centered at the origin. The restriction of f to S^{n} gives rise to an immersion :

w : S^{n} ⟶ C^{n}

∗ Balkan Journal of Geometry and Its Applications, Vol.7, No.2, 2002, pp. 19-26.

\begin{matrix}c \\ ◯ \end{matrix} Balkan Society of Geometers, Geometry Balkan Press 2002.

which has a unique self-intersection point f(−1, 0, ..., 0) = f(1, 0, ..., 0). With respect to the canonical complex structure J on C^{n}, w : S^{n} ⟶ C^{n} is a Lagrangian immersion which is the Whitney immersion.

Let g̃ denote the metric on S^{n} induced from the Euclidien metric on C^{n} via w. We call the Riemannian n−manifold S̃^{n} = (S^{n}, g̃) the Whitney n-sphere. Let S^{n} denote the unit hypersphere of R^{n+1}. Consider the spherical coordinates

{t_{1}, ..., t_{n}} on S^{n} defined by

(1.1)  x_{1} = cos t_{1}, ..., x_{i} = cos t_{i} ∏_{j=1}^{i−1} sin t_{j}, ..., x_{n} = cos t_{n} ∏_{j=1}^{n−1} sin t_{j},

x_{n+1} = sin t_{n} ∏_{j=1}^{n−1} sin t_{j}.

Recall that the Whitney immersion w : S^{n} ⟶ C^{n} is defined by

(1.2)  w(x_{0}, x_{1}, ..., x_{n}) = \frac{1}{1 + x_{0}^{2}} (x_{1}, ..., x_{n}, x_{0}x_{1}, ..., x_{0}x_{n}).

for (x_{0}, x_{1}, ..., x_{n}) ∈ S^{n} and consider the Whitney n−sphere S̃^{n} = (S^{n}, g̃) endowed with the Riemannian metric g̃ induced from the Whitney immersion w. (1.1) and (1.2) imply that the components g̃_{αβ} of the metric tensor g̃ with respect to the spherical coordinates are given by

(1.3)  g̃_{αα} = \frac{∏_{j=1}^{α−1} sin^{2} t_{j}}{1 + cos^{2} t_{1}} , g̃_{αβ} = 0, 1 ≤ α ≠ β ≤ n,

where we put ∏_{i=1}^{0} sin^{2} t_{i} = 1.

Let N and S denote the points (1, 0, ..., 0) and (−1, 0, ..., 0) in S^{n}, respectively. From (1.3) we see that S̃^{n} − \{N, S\} is a warped product (−\frac{π}{2} , \frac{π}{2} ) ×_{ρ(t)} S^{n−1} of the

open interval (−\frac{π}{2} , \frac{π}{2} ) and the unit (n−1)−sphere with warped product metric given by

g̃ = (\frac{1}{1 + cos^{2} t_{1}} ) dt_{1}^{2} + (\frac{sin^{2} t_{1}}{1 + cos^{2} t_{1}} ) g_{0},

where g_{0} is the standard metric on the unit (n − 1)−sphere S^{n−1} and ρ(t) =

\frac{sin t_{1}}{\sqrt{1 + cos^{2} t_{1}}} .

Let {e_{1}, ..., e_{n}} be the unit vector fields in the direction of the tangent vector

fields \{\frac{∂}{∂t_{1}} , ..., \frac{∂}{∂t_{n}} \} on S̃^{n} respectively. Then \{e_{1}, ..., e_{n}, e_{1∗}, ..., e_{n∗}\} form an adapted

Lagrangian orthonormal frame field. By a direct, long computation, we may prove that the second fundamental form of the Whitney immersion w with respect to this adapted frame field satisfies (see [2])

C-Totally Real Submanifolds 21

\begin{aligned}h(e_{1}, e_{1}) & = & 3λe_{1∗}, h(e_{2}, e_{2}) = ... = h(e_{n}, e_{n}) = λe_{1∗}, \\ h(e_{1}, e_{j}) & = & λe_{j∗}, h(e_{j}, e_{k}) = 0, 2 ≤ j ≠ k ≤ n. \end{aligned}

where

λ = −\frac{sin t_{1}}{\sqrt{1 + cos^{2} t_{1}}} .

An orthonormal frame field e_{1}, ..., e_{n}, e_{1∗}, ..., e_{n∗} is called an adapted frame field if e_{1}, ..., e_{n} are orthonormal tangent vector fields and e_{1∗}, ..., e_{n∗} are normal vector fields given by

e_{1∗} = Je_{1}, ..., e_{n∗} = Je_{n}

2 Submanifolds of a Sasakian space form

Let (M̃, g) be a (2m + 1)-dimensional Riemannian manifold endowed with an endomorphism φ ( (1, 1)−tensor field) of its tangent bundle T M̃, a vector field ξ and a 1-form η such that

\{ \begin{matrix}φ^{2}X = −X + η(X)ξ , φξ = 0 , η ∘ φ = 0 , η(ξ) = 1 , \\ g(φX, φY) = g(X, Y) − η(X)η(Y) , η(X) = g(X, ξ) , \end{matrix}

for all vector fields X, Y ∈ Γ(T M̃).

If, in addition, dη(X, Y) = g(φX, Y), then M̃ is said to have a contact Riemannian structure (φ, ξ, η, g). If, moreover, the structure is normal, i.e. if

[φX, φY] + φ^{2}[X, Y] − φ[X, ϕY] − φ[φX, Y] = −2dη(X, Y)ξ,

then the contact Riemannian structure is called a Sasakian structure and M̃ is called a Sasakian manifold. For more details and background, we refer to the standard references [1], [8].

A plane section σ in T_{p}M̃ of a Sasakian manifold M̃ is called a φ-section if it is spanned by X and φX, where X is a unit tangent vector field orthogonal to ξ. The sectional curvature K̄(σ) w.r.t. a φ-section σ is called a φ-sectional curvature. If a Sasakian manifold M̃ has constant φ-sectional curvature c, then it is called a Sasakian space form and is denoted by M̃(c).

The curvature tensor R̃ of a Sasakian space form M̃(c) is given by [1]:

R̃(X, Y)Z = \frac{c + 3}{4}(g(Y, Z)X − g(X, Z)Y)+
+ \frac{c − 1}{4}(η(X)η(Z)Y − η(Y)η(Z)X + g(X, Z)η(Y)ξ − g(Y, Z)η(X)ξ+
+ g(φY, Z)φX − g(φX, Z)φY − 2g(φX, Y)φZ),

for any tangent vector fields X, Y, Z to M̃(c).

An n-dimensional submanifold M of a Sasakian space form M̃(c) is called a C−totally real submanifold if ξ is a normal vector field on M. A direct consequence of this definition is that φ(T M) ⊂ T^{⊥}M, i.e. that M is an anti-invariant submanifold of M̃(c), (hence their name of ”contact”-totally real submanifolds); see e.g. [6].

As examples of Sasakian space forms we mention R^{2m+1} and S^{2m+1}, with standard Sasakian structures.

If M is a Riemannian n−manifold isometrically immersed in a Euclidian m−space E^{m}, one may consider extrinsic invariants as well as intrinsic invariants on M .

Let M be an n-dimensional Riemannian manifold. Denote by K(π) the sectional curvature of the plane section π ⊂ T_{p}M , p ∈ M. For any orthonormal basis {e_{1}, ..., e_{n}} of the tangent space T_{p}M, the scalar curvature τ at p is defined by

τ = ∑_{1≤i<j≤n} K(e_{i} ∧ e_{j}) .

Let p ∈ M and {e_{1}, ..., e_{n}} an orthonormal basis of the tangent space T_{p}M. We denote by H the mean curvature vector, that is

H(p) = \frac{1}{n} ∑_{i=1}^{n} h(e_{i}, e_{i})

Also, we set

h_{ij}^{r} = g(h(e_{i}, e_{j}), e_{r}), ∥h∥^{2} = ∑_{i,j=1}^{n} g(h(e_{i}, e_{j}), h(e_{i}, e_{j})).

3 Main results

Theorem 1. If M^{n} is a C-totally real submanifold of a Sasakian space form M̃^{2n+1}(c), then the mean curvature H and the scalar curvature τ of M satisfy

(3.1)  ∥H∥^{2} ≥ \frac{2(n + 2)}{n^{2}(n − 1)}τ − (\frac{n + 2}{n}) (\frac{c + 3}{4}) .

Moreover the equality sign holds if and only if, with to respect an adapted frame field e_{1}, ..., e_{n}, e_{1∗}, ..., e_{n∗}, e_{2n+1} = ξ with e_{1∗} parallel to H , the second fundamental form of M^{n} in M̃^{2n+1}(c) takes the following form:

h(e_{1}, e_{1}) = 3λe_{1∗}, h(e_{2}, e_{2}) = ... = h(e_{n}, e_{n}) = λe_{1∗},
h(e_{1}, e_{j}) = λe_{j∗} h(e_{j}, e_{k}) = 0, 2 ≤ j ≠ k ≤ n,

with λ ∈ C^{∞}(M). Proof. Let M^{n} be a C-totally real submanifold of a Sasakian space form M̃^{2n+1}(c), and e_{1}, ..., e_{n}, e_{1∗}, ..., e_{n∗}, e_{2n+1} = ξ a local adapted frame field on M^{n}.

Put h_{jk}^{i} = g (h(e_{j}, e_{k}), e_{i∗}).

Then, by

(3.2)  A_{φX}Y = −φh(X, Y) = A_{φY} X ∀X, Y ∈ Γ(T M),

we have

h_{jk}^{i} = h_{ik}^{j} = h_{ij}^{k}, i, j, k = 1, ..., n.

From the definition of the mean curvature function we have

n^{2} ∥H∥^{2} = ∑_{i} (∑_{j} (h_{jj}^{i})^{2} + 2 ∑_{j<k} h_{jj}^{i}h_{kk}^{i}) .

From the equation of Gauss we have

2τ = n(n−1) (\frac{c + 3}{4})+n^{2} ∥H∥^{2}−∥h∥^{2} = n(n−1) (\frac{c + 3}{4})+n^{2} ∥H∥^{2}− ∑_{i,j,k=1}^{n} (h_{jk}^{i})^{2} .

Thus, by applying precedent relations, we obtain

τ = \frac{n(n − 1)}{2} (\frac{c + 3}{4}) + ∑_{i} ∑_{j<k} h_{jj}^{i}h_{kk}^{i} − ∑_{i≠j} (h_{jj}^{i})^{2} − 3 ∑_{i<j<k} (h_{jk}^{i})^{2} .

Let m = \frac{n + 2}{n − 1} . Then, we get

n^{2} ∥H∥^{2} − m (2τ − n(n − 1) (\frac{c + 3}{4})) = ∑_{i} (h_{ii}^{i})^{2} + (1 + 2m) ∑_{i≠j} (h_{jj}^{i})^{2} +

+ 6m ∑_{i<j<k} (h_{jk}^{i})^{2} − 2(m − 1) ∑_{i} ∑_{j<k} h_{jj}^{i}h_{kk}^{i} =

= ∑_{i} (h_{ii}^{i})^{2} + 6m ∑_{i<j<k} (h_{jk}^{i})^{2} + (m − 1) ∑_{i} ∑_{j<k} (h_{jj}^{i} − h_{kk}^{i})^{2} +

+ (1 + 2m − (n − 2)(m − 1)) ∑_{j≠i} (h_{jj}^{i})^{2} − 2(m − 1) ∑_{j≠i} h_{jj}^{i}h_{jj}^{i} =

= 6m ∑_{i<j<k} (h_{jk}^{i})^{2} + (m − 1) ∑_{i≠j,k} ∑_{j<k} (h_{jj}^{i} − h_{kk}^{i})^{2} +

+ \frac{1}{n − 1} ∑_{j≠i} (h_{ii}^{i} − (n − 1)(m − 1)h_{jj}^{i})^{2} ≥ 0

which implies inequality (3.1). We see that the equality sign of (3.1) holds if and only if h_{ii}^{i} = 3h_{jj}^{i}, h_{jk}^{i} = 0, for distinct i, j, k. In particular, if choose e_{1}, ..., e_{n} in such way that φe_{1} is parallel to the mean curvature vector H, we also have h_{kk}^{j} = 0 for j > 1, k = 1, ..., n.

□

Theorem 2. Let i : M^{n} ⟶ R^{2n+1} be a C-totally real isometric immersion satisfying the equality case

(3.3)  ∥H∥^{2} = \frac{2(n + 2)}{n^{2}(n − 1)}τ

Then either M is a totally geodesic submanifold and hence M is locally isometric to the real space R^{n} or the set U of non-totally geodesic points in M is a dense subset of M, U is an open portion of a S̃^{n} Withney sphere with a > 1 and, up to rigid motions of R^{2n+1}, the immersion i is given by w̃, where w̃ : S^{n} ⟶ R^{2n+1} is the immersion lifted from the Whitney immersion.

Proof. It follows from Theorem 1 that the function ϕ = (\frac{n}{n − 2})^{2} ∥H∥^{2} = λ^{2} is a

well-defined function on M. If the function ϕ vanishes identically, then M is a totally geodesic submanifold of R^{2n+1}. So, for simplicity, we may assume from now on that M is non-totally geodesic, i.e. ϕ ≠ 0. Thus, U = {p ∈ M | ϕ(p) ≠ 0} is a non-empty open subset of M.

Let ω^{1}, ..., ω^{n} denote the dual 1-forms of e_{1}, ..., e_{n} and denoted by (ω_{B}^{A}), A, B = 1, ..., n, 1∗, ...n∗, 2n + 1, the connection forms on M defined by

˜∇e_{i} = ∑_{j=1}^{n} ω_{i}^{j}e_{j} + ∑_{j=1}^{n} ω_{i}^{j∗}e_{j∗}, ˜∇e_{i∗} = ∑_{j=1}^{n} ω_{i∗}^{j}e_{j} + ∑_{j=1}^{n} ω_{i∗}^{j∗}e_{j∗}, i = 1, ..., n,

where ω_{i}^{j} = −ω_{j}^{i}, ω_{i∗}^{j∗} = −ω_{j∗}^{i∗} For a C-totally real submanifold M^{n} of a R^{2n+1}, (3.2) yields

ω_{j}^{i∗} = ω_{i}^{j∗}, ω_{i}^{j} = ω_{i∗}^{j∗}, ω_{j}^{i∗} = ∑_{k=1}^{n} h_{jk}^{i}ω^{k}.

We find

(3.4)  ω_{1}^{1∗} = 3λω^{1}, ω_{i}^{1∗} = λω^{i}, ω_{i}^{i∗} = λω^{1}, ω_{j}^{i∗} = 0, 2 ≤ i ≠ j ≤ n.

By applying the equation of Codazzi, we obtain

(3.5)  e_{1}λ = λω_{1}^{2}(e_{2}) = ... = λω_{1}^{n}(e_{n}), e_{2}λ = ... = e_{n}λ = 0,

(3.6)  ω_{1}^{j}(e_{k}) = 0, 1 < j ≠ k ≤ n.

By precedent formulas yield

(3.7)  ω_{1}^{j} = e_{1}(ln λ)ω^{j}, j = 2, ..., n

From Cartan’s structure equations and (3.7) we get dω^{1} = 0 and ∇_{e_{1}}e_{1} = 0.

Therefore, we have the following Lemma 3. On U, the integral curves of φH (or, equivalently, of e_{1}) are geodesics of M.

Let D denote the distribution spanned by φH and D^{⊥} denote the orthogonal complementary distribution of D on U. Then D and D^{⊥} are spanned by {φH} and {e_{2}, ..., e_{n}}, respectively.

By using (3.6) we obtain the following. Lemma 4. On U, the distributions D and D^{⊥} are both integrable. Proof. For any j, k > 1, (3.6) implies

⟨[e_{j}, e_{k}], e_{1}⟩ = ω_{k}^{1}(e_{j}) − ω_{j}^{1}(e_{k}) = 0

Thus, the distribution D^{⊥} is completely integrable. The integrability of D is obvious, since D is a 1-dimensional distribution.

Now, we give the following.

Lemma 5. On U, there exist local coordinate systems {x_{1}, ..., x_{n}} satisfying the following conditions :

(a) D is spanned by \{\frac{∂}{∂x}\} and D^{⊥} is spanned by \{\frac{∂}{∂x_{2}} , ..., \frac{∂}{∂x_{n}} \},

(b) e_{1} = \frac{∂}{∂x} , ω^{1} = dx,

(c) the metric tensor g takes the form : g = dx^{2} + ∑_{j,k=2}^{n} g_{jk}(x, x_{2}, ..., x_{n})dx_{j}dx_{k},

where x = x_{1}. Proof. It is well-know, that there exists a local coordinate systems {y_{1}, ..., y_{n}} such

that e_{1} = \frac{∂}{∂y_{1}} . Since D^{⊥} is completely integrable, there also exists a local coordinate

systems \{z_{1}, ..., z_{n}\} such that \frac{∂}{∂z_{2}} , .., \frac{∂}{∂z_{n}} span D^{⊥}. Put x = x_{1} = y_{1,}x_{2} = z_{2}, ..., x_{n} =

z_{n}, then {x_{1}, ..., x_{n}} is a desired coordinate system.

(3.5) and Lemma 5 imply that λ depends only on x = x_{1}, i.e. λ = λ(x). Let λ′ and λ′′ denote the first and second derivates of λ with respect to x. Lemma 6. On U, the function λ satisfies the following second order ordinary differential equation:

(3.8)  \frac{d^{2}λ}{dx^{2}} + 2λ^{3} = 0

Proof. By taking the exterior differentiation of (3.7) and using (3.4), (3.7) and Cartan’s structure equations, we find

(ln λ)′′ + (ln λ)^{′2} = −2λ^{2}

which is equivalent to (3.8). Lemma 7. The solution of the second order ordinary differential equation (3.8) are given by

(3.9)  λ(x) = −\frac{sin(t(x) + b)}{a\sqrt{1 + cos^{2}(t(x) + b)}},

where t(x) is the inverse function of x(t) defined by

(3.10)  x = ∫_{0}^{t} \frac{adu}{\sqrt{1 + cos^{2}(u + b)}}

and a and b are constants with a > 0 and 0 ≤ b < 2π. Proof. (3.10) implies that x(t) is a strictly increasing differentiable function of t.

Thus, x = x(t) has an inverse function, denoted by t = t(x). From (3.10) we get

(3.11)  \frac{dt}{dx} = \frac{1}{a}\sqrt{1 + cos^{2}(t(x) + b)},

Thus by (3.9), (3.11), and chain rule, we find (3.12)

\frac{dλ}{dx} = −\frac{2 cos(t(x) + b)}{a^{2}(1 + cos^{2}(t(x) + b))}

\frac{d^{2}λ}{dx^{2}} = −\frac{2 sin^{3}(t(x) + b)}{a^{3}(1 + cos^{2}(t(x) + b))^{\frac{3}{2}}}

(3.9) and (3.12) imply that, for any a and b are constants with a > 0 and 0 ≤ b < 2π, the function λ given by (3.9) is a solution of the differential equation (3.8).

Let f = f(x, λ, λ′) = −2λ^{3}. Then f, \frac{∂f}{∂λ}, \frac{∂^{2}f}{∂λ^{2}} are continous functions on the 3-

space R^{3}. Thus, by Existence and Uniquenss Theorem of second ordinary differential equation, the differential equation (3.8) together with the given initial conditions : λ(x_{0}) = λ_{0}, λ′(x_{0}) = λ′_{0}, has a unique solution.

Since for any two arbitrary constants λ_{0}, λ′_{0} we may find real number a and b with a > 0 and 0 ≤ b < 2π which satisfy the following two conditions :

\frac{sin(t(x_{0}) + b)}{a\sqrt{1 + cos^{2}(t(x_{0}) + b)}} = λ_{0}, \frac{2 cos(t(x_{0}) + b)}{a^{2}(1 + cos^{2}(t(x_{0}) + b)} = λ′_{0},

therefore every solution of the differential equation (3.8) takes the form given by (3.10). The rigidy theorem of C-totally real immersions in R^{2n+1} achieves the proof.

□

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University Politehnica of Bucharest, Department of Mathematics I

Splaiul Independenţei 313, 77206 Bucharest, Romania