Hypersurfaces with Two Distinct Para-Blaschke Eigenvalues in 𝑆^𝑛+1(1)

Abstract

Let be an -dimensional immersed hypersurface without umbilical points and with vanishing Möbius form in a unit sphere , and let and be the Blaschke tensor and the Möbius second fundamental form of , respectively. We define a symmetric tensor which is called the para-Blaschke tensor of , where is a constant. An eigenvalue of the para-Blaschke tensor is called a para-Blaschke eigenvalue of . The aim of this paper is to classify the oriented hypersurfaces in with two distinct para-Blaschke eigenvalues under some rigidity conditions.

1. Introduction

In the Möbius geometry of hypersurfaces, Wang [1] studied invariants of hypersurfaces in a unit sphere under the Möbius transformation group. Let be an -dimensional immersed hypersurface without umbilical points in . We choose a local orthonormal basis for the induced metric with dual basis . Let be the second fundamental form of the immersion and the mean curvature of the immersion . By putting , the Möbius metric of the immersion is defined by which is a Möbius invariant. , , and are called the Möbius form, the Blaschke tensor, and the Möbius second fundamental formof the immersion , respectively (see [1]), where and and are the Hessian matrix and the gradient with respect to the induced metric . It was proved by [1] that , , and are Möbius invariants.

In the study of the Möbius geometry of hypersurfaces, one of the important aims is to characterize hypersurfaces in terms of Möbius invariants. Concerning this topic, there are many important results; one can see [2–9]. We should notice that [5] classified all umbilic-free hypersurfaces with parallel Möbius second fundamental form. Recently, by making use of the two important Möbius invariants, the Blaschke tensor and the Möbius second fundamental form of , Zhong and Sun [10] defined a symmetric tensor which is called the para-Blaschke tensor of , where is a constant. An eigenvalue of the para-Blaschke tensor is called a para-Blaschke eigenvalue of . In [7], Li and Wang investigated and completely classified hypersurfaces without umbilical points and with vanishing Möbius form in , which satisfy . It should be noted that the condition implies that the para-Blaschke eigenvalues of are all equal. If has two distinct constant para-Blaschke eigenvalues, Zhong and Sun [10] obtained the following.

Theorem 1 (see [10]). Let be an -dimensional immersed hypersurface without umbilical points. If is of two distinct constant para-Blaschke eigenvalues and of vanishing Möbius form , then is locally Möbius equivalent to(1)the Riemannian product in , or(2)the image of of the standard cylinder in , or(3)the image of of the Riemannian product in , or(4)the hypersurface as indicated in Example 11, or(5)the hypersurface as indicated in Example 12.

Remark 2. In Theorem 1 and the following Theorems, the conformal diffeomorphisms and are defined by respectively, where is the open hemisphere in whose first coordinate is positive (see [1]).

In this paper, we study hypersurfaces without umbilical points and with vanishing Möbius form. If is of two distinct para-Blaschke eigenvalues, we obtain some classification Theorems in terms of some rigidity conditions.

Theorem 3. Let be an   -dimensional immersed hypersurface without umbilical points and with vanishing Möbius form and constant Möbius scalar curvature in . If is of two distinct para-Blaschke eigenvalues and the multiplicities of these two distinct para-Blaschke eigenvalues are greater than 1, then is locally Möbius equivalent to(1)the Riemannian product in , or(2)the image of of the standard cylinder in , or(3)the image of of the Riemannian product in , or(4)the hypersurface as indicated in Example 11, or(5)the hypersurface as indicated in Example 12, where .

Theorem 4. Let be an   -dimensional immersed complete hypersurface without umbilical points and with vanishing Möbius form and constant Möbius scalar curvature in . If is of two distinct para-Blaschke eigenvalues one of which is simple, then we have the following.(1)If the Möbius sectional curvature of is nonnegative and , then is locally Möbius equivalent to(i)the Riemannian product in , or(ii)the image of of the standard cylinder in .(2)If  then is locally Möbius equivalent to: (i)the Riemannian product in , or(ii)the image of of the standard cylinder in , in this case , or(iii)the image of of the Riemannian product in , in this case , where .

If , then the para-Blaschke tensor is the Blaschke tensor of . From Theorems 3 and 4, we have the following.

Corollary 5. Let be an   -dimensional immersed hypersurface without umbilical points and with vanishing Möbius form and constant Möbius scalar curvature in . If is of two distinct Blaschke eigenvalues and the multiplicities of these two distinct Blaschke eigenvalues are greater than 1, then is locally Möbius equivalent to(1)the Riemannian product in , or(2)the image of of the standard cylinder in , or(3)the image of of the Riemannian product in , or(4)the minimal hypersurface as indicated in Example 13, or(5)the nonminimal hypersurface as indicated in Example 14, where .

Corollary 6. Let be an   -dimensional immersed complete hypersurface without umbilical points and with vanishing Möbius form and constant Möbius scalar curvature in . If is of two distinct Blaschke eigenvalues one of which is simple, then we have the following.(1)If the Möbius sectional curvature of is nonnegative and , then is locally Möbius equivalent to:(i)the Riemannian product in , or(ii)the image of of the standard cylinder in .(2)If  then is locally Möbius equivalent to(i)the Riemannian product in , or(ii)the image of of the standard cylinder in , or(iii)the image of of the Riemannian product in , where .

2. Preliminaries

In this section, we review the definitions of Möbius invariants and the fundamental formulas on Möbius geometry of hypersurfaces in ; for more details see [1].

Let be an -dimensional hypersurface of without umbilical points. We use the following range of indices throughout this paper: The structure equations on with respect to the Möbius metric can be written as follows: where are -forms on with By exterior differentiation of these equations, we get where and , , and are locally defined functions and satisfy (1), (2), and (3). We have Let , , be the covariant derivative of , , . From the structure equations (8), we infer where denotes the curvature tensor with respect to the Möbius metric on and is the Möbius scalar curvature of the immersion .

Since the Möbius form , by (19)–(21), we have for all indices , , that

Denote by the para-Blaschke tensor; then, where is constant. The covariant derivative of is defined by From (24), we have From (23), we have for all indices , , that

3. Lemma and Examples

In this section, we make the following convention on the ranges of indices: The following Lemma 7 due to Li et al. [6] will be used.

Lemma 7 (see [6]). Let be an -dimensional hypersurface with two distinct principal curvatures with multiplicities and , respectively. Then, the principal curvatures of the Möbius second fundamental form of are constant, which are given by

In [4], Cheng and Shu studied the Möbius invariants on some typical examples.

Example 8 (see [4]). Let and be the standard embedding of the unit spheres. Then, the Riemannian product defined by , for any and any , is an immersed hypersurface without umbilical points and with vanishing Möbius form in . By (3.4) in [4], we know that the Möbius sectional curvature of is nonnegative. From (3.9) to (3.11) in [4], Lemma 7, and (24) and by a simple and direct calculation, we see that Thus, has exactly two distinct constant para-Blaschke eigenvalues. Defining the trace-free tensor , we see that or Thus, we know that , if and only if or , and , if and only if or .

Example 9 (see [4]). Let be the standard cylinder. Then, the hypersurface is an immersed hypersurface without umbilical points and with vanishing Möbius form in . By (3.25) in [4], we know that the Möbius sectional curvature of is nonnegative. From (3.31)–(3.33) in [4], Lemma 7 and (24), we can calculate that Thus has exactly two distinct constant para-Blaschke eigenvalues. By the similar calculation in Example 8, we have We know that if and only if . If , from (3.23) in [4], we have .

Example 10 (see [4]). Let be the standard embedding. Then the hypersurface is an immersed hypersurface without umbilical points and with vanishing Möbius form in . By (3.41) in [4], we know that the Möbius sectional curvature of is not nonnegative. From (3.45)–(3.47) in [4], Lemma 7 and (24), we can calculate that Thus, has exactly two distinct constant para-Blaschke eigenvalues. By the similar calculation in Example 8, we have We know that if and only if . If , from (3.39) in [4], we have .

Recently, Zhong and Sun [10] studied some important examples, which can be stated as follows.

Example 11 (see [10]). Hypersurface , where ,  ,  , , , ,  . Here, is an immersed hypersurface in -dimensional sphere , which is at least of two nonzero principal curvatures and with constant mean curvature and constant scalar curvature : and is the canonical embedding of -dimensional hyperbolic space to -dimensional Lorentzian space and ,  . By [10], we know that the Möbius sectional curvature of is not nonnegative. From (3.12) to (3.21) in [10], by a simple and direct calculation, we see that the para-Blaschke tensor of is Clearly, the Möbius form and has exactly two distinct constant para-Blaschke eigenvalues. We can easily check that if or .

Example 12 (see [10]). Hypersurface , where ,  ,  , ,  ,  ,  . Here, is an immersed hypersurface in -dimensional hyperbolic space , which is at least of two nonzero principal curvatures and with constant mean curvature and constant scalar curvature : and , and is the canonical embedding of -dimensional sphere to -dimensional Euclidean space . By [10], we know that the Möbius sectional curvature of is not nonnegative. From (3.39) to (3.48) in [10], by a simple and direct calculation, we see that the para-Blaschke tensor of is Clearly, the Möbius form and has exactly two distinct constant para-Blaschke eigenvalues. We can easily check that if or .

If , Examples 11 and 12 decrease to the following Examples (see [8]).

Example 13. Minimal hypersurface , where , , , , , , . Here, is an immersed minimal hypersurface without umbilical points in -dimensional sphere and with constant scalar curvature : and is the canonical embedding of -dimensional hyperbolic space to -dimensional Lorentzian space .

Example 14. Nonminimal hypersurface , where , , , , , , . Here, is an immersed minimal hypersurface without umbilical points in -dimensional hyperbolic space and with constant scalar curvature : and is the canonical embedding of -dimensional sphere to -dimensional Euclidean space .

4. Proof of Theorems

Let , , and denote the -symmetric matrices , , and , respectively, where , , and are defined by (1), (2), and (24). Since from (23), we know that , , and hold, we can choose a local orthonormal basis such that , , and , where , , and are the Blaschke eigenvalues, the Möbius principal curvatures, and the para-Blaschke eigenvalues of the immersion .

Since the Möbius form is zero, from (27), we know that the para-Blaschke tensor is Codazzi tensor. By the similar proof of Proposition  3.1 in [11], we have the following.

Proposition 15. Let be an -dimensional hypersurface with vanishing Möbius form in a unit sphere . If the multiplicity of a para-Blaschke eigenvalue is constant and greater than , then this para-Blaschke eigenvalue is constant along its leaf.

Proof of Theorem 3. Let be an immersed hypersurface without umbilical points and with vanishing Möbius form and constant Möbius scalar curvature in . Denote by the para-Blaschke eigenvalues of multiplicities and , respectively, where . We have from (18) and (24) that Denote by and the leaf of the para-Blaschke eigenvalues and , respectively. From Proposition 15, we know that is constant on . Since the Möbius scalar curvature is constant, (44) implies that is constant on . By Proposition 15, again, is constant on . Therefore, we know that is constant on . By the same assertion, we know that is constant on . Therefore has two distinct constant para-Blaschke eigenvalues. By Theorem 1, we know that Theorem 3 is proved.

Next, we consider the case that is of two distinct para-Blaschke eigenvalues, one of which is simple. We assume that is of vanishing Möbius form and constant Möbius scalar curvature . Without loss of generality, we may assume where for are the para-Blaschke eigenvalues of . Therefore, we know that Since , we have .

We choose a local orthonormal frame such that is a unit principal vector with respect to . From (25), we have where is the Levi-Civita connection for the Möbius metric given by

From (27) and (48), we easily see that where , . From (50) and the formula we have Therefore, from (11) we have We may consider locally that is a function of the arc length of the integral curve of the principal vector field corresponding to the principal curvature . Therefore, we may put . Thus, for , we get and for From (17), (22), and (55), we have for that From (55), we have for that From the above two equalities, we have for We define and obtain from the above equation that

On the other hand, from (55), we have . By the definition of geodesic, we know that any integral curve of the principal vector field corresponding to the principal curvature is a geodesic. Thus, we see that is a function defined in since is complete and any integral curve of the principal vector field corresponding to is a geodesic.

We can prove the following.

Proposition 16. If , then the positive function is bounded.

Proof. By (59) and taking summation over , , we have that is, from (18) From the definition of and (46), we see that for , or for .
From (18), we have . Thus, we have . We infer that From (63) and (64), we have for , or for .
Since is existing, we know that for some fixed , and are two constants. Multiplying (65) and (66) by , we have two cases.(i)If , for any , integrating from to , then  for , or  for .(ii)If , for any , integrating from to , then  for , or  for .
For the two cases above, we all have for , or for .
Therefore, we have for , or for .
Since , from (73) and (74), we see that the positive function is bounded from above. Proposition 16 is proved.

Proof of Theorem 4. (1) Since the Möbius sectional curvature of is nonnegative, from (22), we have for . Therefore, we know that , . From (59), we have . Thus, is a monotonic function of . Therefore, as observed by Wei [12], must be monotonic when tends to infinity. Since we assume that , from Proposition 16, we know that the positive function is bounded. Since is bounded and monotonic when tends to infinity, we know that both and exist and then we get By the monotonicity of , we have and . From , we know that is constant, and therefore is also constant from (46). We obtain that has two distinct constant para-Blaschke eigenvalues. From Theorem 1 and Examples 8–12, we know that the result (1) of Theorem 4 is proved.
(2) From (46) and (47), we have Putting , we have . Therefore, (76) can be rewritten as We have from (77) From (46), we obtain or If there exists a point on such that (79) and (80) hold at ; that is, we have at . From , we have at ; this is in contradiction to the assumption that has two distinct para-Blaschke eigenvalues. Therefore, we only consider two cases.
Case (i). If (79) holds on , by (64) and (5), we have Case (ii). If (80) holds on , by (64) and (5), we have From (61), (81), and (82), we all have . Thus, is a monotonic function of . Therefore, we have the following: must be monotonic when tends to infinity. Since and , we have . From Proposition 16, we know that the positive function is bounded. Therefore, by the similar assertion in the proof of (1), we obtain that has two distinct constant para-Blaschke eigenvalues. From Theorem 1 and Examples 8–12, we know that Theorem 4 is proved.

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referee for his valuable remarks and suggestions that really improve the paper. Thanks are also due to Shaanxi Education Department (11JK0479).