The Global Property of Generic Conformally Flat Hypersurfaces in ${\mathbb{R}}^4$

Abstract

A conformally flat hypersurface $f:M^3\to {\mathbb{R}}^4$ in the four-dimensional Euclidean space ${\mathbb{R}}^4$ is said to be generic if the hypersurface has three distinct principal curvatures everywhere. In this paper, we study the generic conformally flat hypersurfaces in ${\mathbb{R}}^4$ using the framework of Möbius geometry. First, we classify locally the generic conformally flat hypersurfaces with a vanishing Möbius form under the Möbius transformation group of ${\mathbb{R}}^4$. Second, we investigate the global behavior of the compact generic conformally flat hypersurfaces and give some integral formulas about the Möbius invariant of these hypersurfaces.

1. Introduction

A Riemannian manifold $(M^n,g)$ is conformally flat if every point has a neighborhood which is conformal to an open set in the Euclidean space ${\mathbb{R}}^n$. A hypersurface in the Euclidean space ${\mathbb{R}}^{n+1}$ is said to be conformally flat if it is with respect to the induced metric. The dimension of the hypersurface seems to play an important role in the study of conformally flat hypersurfaces. For $n\ge 4$, the immersed hypersurface $f:M^n\to {\mathbb{R}}^{n+1}$ is conformally flat if and only if at least $n-1$ of the principal curvatures coincide at each point according to the result of Cartan-Schouten [1,2]. Cartan-Schouten’s result is no longer true for three-dimensional hypersurfaces. In fact, the number of distinct principal curvatures of a three-dimensional conformally flat hypersurface may be $1,2$, or 3. Lancaster [3] gave some examples of conformally flat hypersurfaces in ${\mathbb{R}}^4$ having three different principal curvatures. These standard examples come from cones, cylinders, or rotational hypersurfaces over a surface with a constant Gaussian curvature in three-sphere ${\mathbb{S}}^3$, Euclidean three-space ${\mathbb{R}}^3$, or hyperbolic three-space ${\mathbb{R}}_{+}^3$, respectively. For $n=2$, the existence of isothermal coordinates means that any Riemannian surface is conformally flat.

In [1], E. Cartan gave a local classification for conformally flat hypersurfaces for dimensions $n\ge 4$, and Cartan proved that if $f:M^n\to R^{n+1}$, then $n\ge 4$ is a conformally flat hypersurface if and only if f is a branched channel hypersurface (see [1], Section 24). In [4], Do Carmo, Dajczer, and Mercuri studied the diffeomorphism types of the compact conformally flat hypersurfaces of dimensions $n\ge 4$. Pinkall in [5] studied the intrinsic conformal geometry of compact conformally flat hypersurfaces. Suyama in [6] explicitly constructed compact conformally flat hypersurfaces in space forms using co-dimension single foliation with $(n-1)$ spheres.

If a conformally flat hypersurface $f:M^3\to {\mathbb{R}}^4$ in ${\mathbb{R}}^4$ has two distinct principal curvatures, then it shares the same geometric properties with the conformally flat hypersurfaces of dimensions $n\ge 4$. For a generic conformally flat hypersurface $f:M^3\to {\mathbb{R}}^4$, according to the theorem in [1], there exists an orthogonal curvature line coordinate system at each point of such hypersurfaces. In a series of papers [7,8,9,10], Jeromin and Suyama studied Guichard’s nets on generic conformally flat hypersurfaces. In other papers [6,11,12], Suyama locally classified the generic conformally flat hypersurfaces in terms of the first fundamental form. The (local) classification of these hypersurfaces is far from complete. For generic conformally flat hypersurfaces, whether all principal curvatures are distinct is an open condition, and thus the geometry of generic conformally flat hypersurfaces is more fruitful.

It is known that the conformal transformation group of ${\mathbb{R}}^n$ is isomorphic to its Möbius transformation group if $n\ge 3$. As conformal invariant objects, the Möbius geometry is the most suitable framework for the study of conformally flat hypersurfaces. The Möbius form C is an important invariant in the Möbius geometry of hypersurfaces, which determines whether the Möbius second fundamental form is a Codazzi tensor. For a hypersurface $x:M^n\to {\mathbb{R}}^{n+1}$, let $\{e_1,\cdots ,e_n\}$ be an orthonormal basis with respect to the induced metric $I=df·df$ with the dual basis $\left\{{\theta }_i\right\}$, and let $II={\sum }_{ij}{h}_{ij}{\theta }_i{\theta }_j$, and $H={\sum }_i\frac{h_{ii}}{n}$ be the second fundamental form and the mean curvature of f, respectively. Let ${\rho }^2=\frac{n}{n-1}{\left(\right|II|}^2-n{H}^2)$. Then, the Möbius form C is defined as follows:

$$C=-{\rho }^{-1}\sum _i[e_i\left(H\right)+\sum _j(h_{ij}-H{\delta }_{ij})e_j(log\rho )]{\theta }_i,$$

In [13], Li Haizhong and Wang Changping classified the surfaces with vanishing-Möbius forms (alternatively, see [14]). Guo Zhen and Li Fengjiang studied surfaces with closed Möbius forms in [15]. For high-dimensional hypersurfaces, Lin Limiao and Guo zhen classified the hypersurfaces with two or three constant Möbius principal curvatures and a closed Möbius form in [16,17]. In the study of other special hypersurfaces, the Möbius form vanishing is a necessary condition (see [17,18,19]). In [16], Lin and Guo classified the conformally flat hypersurface of dimensions $n\ge 4$ with a closed Möbius form in Möbius geometry. Lin and Guo’s results are still valid for three-dimensional conformally flat hypersurfaces in ${\mathbb{R}}^4$ if the number of distinct principal curvatures of the hypersurfaces is two. In this paper, using the framework of Möbius geometry, we investigate the generic conformally flat hypersurfaces in ${\mathbb{R}}^4$. First, we classify the generic conformally flat hypersurfaces with a vanishing Möbius form:

Theorem 1.

Let $f:M^3\to {\mathbb{R}}^4$ be a generic conformally flat hypersurface. If the Möbius form vanishes, then f is locally Möbius equivalent to a cone over a homogeneous torus $x:{\mathbb{S}}^1\left(r\right)\times {\mathbb{S}}^1\left(\sqrt{1-r^2}\right)\to {\mathbb{S}}^3\subset {\mathbb{R}}^4$ in three-sphere ${\mathbb{S}}^3$.

Let $f:M^3\to {\mathbb{R}}^4$ be an immersed hypersurface with the three distinct principal curvatures ${\lambda }_1,{\lambda }_2$, and ${\lambda }_3$. Then, the Möbius curvature

$${\mathbb{M}}_{123}=\frac{{\lambda }_1-{\lambda }_2}{{\lambda }_1-{\lambda }_3}$$

is only an extrinsic Möbius invariant. If the Möbius curvature of a generic conformally flat hypersurface is constant, then we prove that the Möbius form vanishes, and thus we have the following results:

Proposition 1.

Let $f:M^3\to {\mathbb{R}}^4$ be a generic conformally flat hypersurface. If the Möbius curvature $M_{123}$ is constant, then f is locally Möbius equivalent to a cone over a homogeneous torus $x:{\mathbb{S}}^1\left(r\right)\times {\mathbb{S}}^1\left(\sqrt{1-r^2}\right)\to {\mathbb{S}}^3\subset {\mathbb{R}}^4$ in three-sphere ${\mathbb{S}}^3$.

Second, we investigate the global behavior of compact generic conformally flat hypersurfaces with Möbius invariants. The Ricci curvature of the hypersurface is called nonpositive pointwise if the Ricci curvature $Ri{c}_p\le 0$ for all $p\in M^n$. Similarly, a nonnegative pointwise Ricci curvature means that $Ri{c}_p\ge 0$ for all $p\in M^n$:

Theorem 2.

Let $f:M^3\to {\mathbb{R}}^4$ be a generic conformally flat hypersurface. If the hypersurface $M^3$ is compact, then the Ricci curvature of the Möbius metric cannot be nonpositive pointwise or nonnegative pointwise.

Theorem 3.

Let $f:M^3\to {\mathbb{R}}^4$ be a generic conformally flat hypersurface and g and A be the Möbius metric and the Blaschke tensor, respectively. Let $\tilde{A}:=A-\frac{1}{3}tr\left(A\right)g$. If the hypersurface $M^3$ is compact, then

$${\int }_{M^3}\left[|\tilde{A}{|}^2+\frac{1}{3}{R}^2-\frac{4}{9}R-{\left|Ric\right|}^2-\frac{2}{27}\right]d{v}_g=0,$$

where $\left|Ric\right|$ and R denote the norm of the Ricci curvature and the scalar curvature of the Möbius metric g, respectively.

Corollary 1.

Let $f:M^3\to {\mathbb{R}}^4$ be a generic conformally flat hypersurface and g be its Möbius metric. If the hypersurface $M^3$ is compact, then

$${\int }_{M^3}\left[\left|Ric\right|-\frac{2\sqrt{2}}{3}R\right]d{v}_g>0.$$

Corollary 2.

Let $f:M^3\to {\mathbb{R}}^4$ be a generic conformally flat hypersurface and g be its Möbius metric. If the hypersurface $M^3$ is compact, then

$${\int }_{M^3}\left[|\tilde{A}{|}^2-\frac{16}{9}tr\left(A\right)+\frac{2}{9}\right]d{v}_g>0.$$

Remark 1.

There exist many examples of generic conformally flat hypersurfaces in ${\mathbb{R}}^4$ (see [6,7,8,11] or Section 3), but these examples are local and noncompact. It is difficult to build a compact, generic conformally flat hypersurface with these local examples. Theorems 2 and 3 show that there are many obstructions to constructing a compact, generic conformally flat hypersurface in ${\mathbb{R}}^4$, and it is relatively rare.

Remark 2.

Due to conformal invariant objects, the theory of conformally flat hypersurfaces is essentially the same whether it is considered in the space forms ${\mathbb{R}}^{n+1}$, ${\mathbb{S}}^{n+1}$, or ${\mathbb{H}}^{n+1}$. In fact, there exists conformal diffeomorphism between the space forms. The $(n+1)$-dimensional hyperbolic space ${\mathbb{H}}^{n+1}$ defined by

$${\mathbb{H}}^{n+1}=\left\{(y_0,y_1,\cdots ,y_{n+1})\right|-y_0^2+y_1^2+\cdots +y_{n+1}^2=-1,y_0>0\}.$$

The conformal diffeomorphisms σ and τ are defined by

$$\begin{array}{cc}\hfill & \sigma :{\mathbb{R}}^{n+1}\to {\mathbb{S}}^{n+1}\backslash \{(-1,\overrightarrow{0})\},\sigma \left(u\right)=(\frac{1-{\left|u\right|}^2}{1+{\left|u\right|}^2},\frac{2u}{1+{\left|u\right|}^2}),\hfill \\ \hfill & \tau :{\mathbb{H}}^{n+1}\to {\mathbb{S}}_{+}^{n+1}\subset {\mathbb{S}}^{n+1},\tau \left(y\right)=(\frac{1}{y_0},\frac{\overrightarrow{y}}{y_0}),y=(y_0,\overrightarrow{y})\in {\mathbb{H}}^{n+1},\hfill \end{array}$$

where ${\mathbb{S}}_{+}^{n+1}$ is the hemisphere in ${\mathbb{S}}^{n+1}$, in which the first coordinate is positive. Under conformal diffeomorphisms σ and τ, the conformally flat hypersurfaces in the space forms are equivalent to each other.

This paper is organized as follows. In Section 2, we review the elementary facts about the Möbius geometry of hypersurfaces in ${\mathbb{R}}^{n+1}$. In Section 3, we investigate the Möbius invariants of generic conformally flat hypersurfaces in ${\mathbb{R}}^4$ and prove Theorem 1 and Proposition 1. In Section 4, we investigate the global behavior of compact generic conformally flat hypersurfaces in ${\mathbb{R}}^4$ and prove Theorems 2 and 3.

2. Möbius Invariants of Hypersurfaces in ${\mathbb{R}}^{\mathbf{n}+\mathbf{1}}$

In this section, we define Möbius invariants and give a congruent theorem of hypersurfaces in ${\mathbb{R}}^{n+1}$. For details, we refer the reader to [20,21].

Let ${\mathbb{R}}^{n+2}$ denote the $(n+2)$-dimensional Euclidean space and a dot · represent its inner product. The $(n+1)$-dimensional sphere is ${\mathbb{S}}^{n+1}=\{x\in {\mathbb{R}}^{n+2}|x·x=1\}$. The hypersphere $S_p\left(\rho \right)$ in ${\mathbb{S}}^{n+1}$ with acenter $p\in {\mathbb{S}}^{n+1}$ and radius $\rho$ is given by

$$S_p\left(\rho \right)=\{y\in {\mathbb{S}}^{n+1}|p·y=cos\rho \},0<\rho <\pi .$$

Let ${\mathbb{D}}^{n+2}=\{x\in {\mathbb{R}}^{n+2}|x·x\le 1\}$. By taking $o\in {\mathbb{R}}^{n+2}$ such that $o\notin {\mathbb{D}}^{n+2}$, a line l that passes through the point o intersects the sphere ${\mathbb{S}}^{n+1}$ in two points $p,q$. Now we define the Möbius inversion ${\mathrm{Y}}_o$ for the point $o\notin {\mathbb{D}}^{n+2}\subset {\mathbb{R}}^{n+2}$ as follows:

$${\mathrm{Y}}_o:{\mathbb{S}}^{n+1}\to {\mathbb{S}}^{n+1},{\mathrm{Y}}_o\left(p\right)=q.$$

Clearly, ${\mathrm{Y}}_o\in \mathbb{M}\left({\mathbb{S}}^{n+1}\right)$. When the point o is at infinity, the Möbius inversion is indeed a reflection ${\mathrm{Y}}_o\in \mathrm{O}(n+2)$ and thus an isometric transformation of ${\mathbb{S}}^{n+1}$:

Proposition 2

([22]). The Möbius transformation group $\mathbb{M}\left({\mathbb{S}}^{n+1}\right)$ is generated by Möbius inversions ${\mathrm{Y}}_o$.

Let ${\mathbb{R}}_1^{n+3}$ be the Lorentz space (i.e., ${\mathbb{R}}^{n+3}$), with the scalar product $\langle ,\rangle$ defined by

$$\langle x,y\rangle =-x_0{y}_0+x_1{y}_1+\cdots +x_{n+2}{y}_{n+2}$$

for $x=(x_0,x_1,\cdots ,x_{n+2}),y=(y_0,y_1,\cdots ,y_{n+2})\in {\mathbb{R}}^{n+3}$.

Let $GL\left({\mathbb{R}}^{n+3}\right)$ be the set of an invertible $(n+3)\times (n+3)$ matrix. Then, the Lorentz orthogonal group $\mathrm{O}(n+2,1)$ is defined by

$$\mathrm{O}(n+2,1)=\{T\in GL\left({\mathbb{R}}^{n+3}\right)|T{I}_1{T}^t=I_1\},$$

where $T^t$ denotes the transpose of the matrix T, $I_1=\left(\begin{array}{cc}-1& 0\\ 0& I\end{array}\right),$ and I is the $(n+2)\times (n+2)$ unit matrix.

The positive light cone is

$${\mathbb{C}}_{+}^{n+2}=\{y=(y_0,{\overrightarrow{y}}_1)\in \mathbb{R}\times {\mathbb{R}}^{n+2}={\mathbb{R}}_1^{n+3}|\langle y,y\rangle =0,y_0>0\},$$

and ${\mathrm{O}}^{+}(n+2,1)$ is the subgroup of $\mathrm{O}(n+2,1)$ defined by

$${\mathrm{O}}^{+}(n+2,1)=\{T\in \mathrm{O}(n+2,1)|T\left(C_{+}^{n+2}\right)=C_{+}^{n+2}\}.$$

Proposition 3

([23]). Let $T=\left(\begin{array}{cc}w& u\\ v& Q\end{array}\right)\in \mathrm{O}(n+2,1)$, where Q is an $(n+2)\times (n+2)$ matrix. Then, $T\in {\mathrm{O}}^{+}(n+2,1)$ if and only if $w>0.$

It is well known that the subgroup ${\mathrm{O}}^{+}(n+2,1)$ is isomorphic to the Möbius transformation group $\mathbb{M}\left({\mathbb{S}}^{n+1}\right)$. In fact, for any

$$T=\left(\begin{array}{cc}w& u\\ v& Q\end{array}\right)\in {\mathrm{O}}^{+}(n+2,1),$$

we can define the Möbius transformation $\mathsf{\Psi }\left(T\right):{\mathbb{S}}^{n+1}\mapsto {\mathbb{S}}^{n+1}$ as

$$\mathsf{\Psi }\left(T\right)\left(x\right)=\frac{Qx+v}{ux+w},x={(x_1,\cdots ,x_{n+2})}^t\in {\mathbb{S}}^{n+1}.$$

Then the map $\mathsf{\Psi }:{\mathrm{O}}^{+}(n+2,1)\mapsto \mathbb{M}\left({\mathbb{S}}^{n+1}\right)$ is a group isomorphism.

Let $Q\in \mathrm{O}(n+2)$ be an isometric transformation of ${\mathbb{S}}^{n+1}$. Then, $Q\in \mathbb{M}\left({\mathbb{S}}^{n+1}\right)$, and

$${\mathsf{\Psi }}^{-1}\left(Q\right)=\left(\begin{array}{cc}1& 0\\ 0& Q\end{array}\right)\in {\mathrm{O}}^{+}(n+2,1).$$

Thus, ${\mathsf{\Psi }}^{-1}\left(\mathrm{O}(n+2)\right)\subset {\mathrm{O}}^{+}(n+2,1)$ is a subgroup.

The $(n+1)$-dimensional sphere ${\mathbb{S}}^{n+1}$ is diffeomorphic to the projective light cone $P{C}^{n+1}$ such that

$$P{C}^{n+1}=\{\left[z\right]\in P{R}^{n+3}|z\in {\mathbb{C}}_{+}^{n+2}\}.$$

The diffeomorphism $\mathsf{\Phi }:{\mathbb{S}}^{n+1}\to P{C}^{n+1}$ is given by

$$\mathsf{\Phi }\left(x\right)=\left[y\right]=\left[\right(1,x\left)\right].$$

The group ${\mathrm{O}}^{+}(n+2,1)$ acts on $P{C}^{n+1}$ as follows:

$$T\left[p\right]=\left[Tp\right],T\in {\mathrm{O}}^{+}(n+2,1),\left[p\right]\in P{C}^{n+1}.$$

With conformal map $\sigma :{\mathbb{R}}^{n+1}\to {\mathbb{S}}^{n+1}\backslash \{(-1,\overrightarrow{0})\},$ we can obtain the conformal invariants of hypersurfaces in ${\mathbb{R}}^{n+1}$. Let $f:M^n\to {\mathbb{R}}^{n+1}$ be a hypersurface without umbilical points and $\left\{e_i\right\}$ be an orthonormal basis with respect to the induced metric $I=df·df$ with the dual basis $\left\{{\theta }_i\right\}$. Let $II={\sum }_{ij}{h}_{ij}{\theta }_i{\theta }_j$ and $H={\sum }_i\frac{h_{ii}}{n}$ be the second fundamental form and the mean curvature of f, respectively. To study the Möbius geometry of f, as in [20,21], one considers the Möbius position vector $Y:M^n\to {\mathbb{R}}_1^{n+3}$ of f as follows:

$$Y=\rho \left(\frac{1+f·f}{2},\frac{1-f·f}{2},f\right),{\rho }^2=\frac{n}{n-1}{\left(\right|II|}^2-n{H}^2),$$

where · denotes the Euclidean inner product in ${\mathbb{R}}^{n+1}$.

Theorem 4

([21]). Two hypersurfaces $f,\overline{f}:M^n\to {\mathbb{R}}^{n+1}$ are Möbius equivalent if and only if there exists T in the Lorentz group $O(n+2,1)$ such that $\overline{Y}=YT.$

It follows immediately from Theorem 4 that

$$g=\langle dY,dY\rangle ={\rho }^{2}df·df$$

is a Möbius invariant called the Möbius metric of f.

Let $\Delta$ be the Laplacian with respect to g. We define

$$N=-\frac{1}{n}\Delta Y-\frac{1}{2n^2}<\Delta Y,\Delta Y>Y,$$

which satisfies $\langle Y,Y\rangle =0=\langle N,N\rangle ,\langle N,Y\rangle =1.$

Let $\{E_1,\cdots ,E_n\}$ be a local orthonormal basis for $(M^n,g)$ with a dual basis $\{{\omega }_1,\cdots ,{\omega }_n\}$. We write $Y_i=E_i\left(Y\right)$, and then we have

$$\langle Y_i,Y\rangle =\langle Y_i,N\rangle =0,\langle Y_i,Y_j\rangle ={\delta }_{ij},1\le i,j\le n.$$

Let $\xi$ be the mean curvature sphere of f, written as

$$\xi =\left(\frac{1+{\left|f\right|}^2}{2}H+f·e_{n+1},\frac{1-{\left|f\right|}^2}{2}H-f·e_{n+1},Hf+e_{n+1}\right),$$

where $e_{n+1}$ is the unit normal vector field of f in ${\mathbb{R}}^{n+1}$. Thus, $\{Y,N,Y_1,\cdots ,Y_n,\xi \}$ forms a moving frame in ${\mathbb{R}}_1^{n+3}$ along $M^n$. We will use the following range of indices in this section: $1\le i,j,k\le n$. We can write the structure equations as follows:

$$\begin{array}{ccc}& & dY=\sum _i{Y}_i{\omega }_i,\hfill \\ & & dN=\sum _{ij}{A}_{ij}{\omega }_i{Y}_j+\sum _i{C}_i{\omega }_i\xi ,\hfill \\ & & d{Y}_i=-\sum _j{A}_{ij}{\omega }_{j}Y-{\omega }_{i}N+\sum _j{\omega }_{ij}{Y}_j+\sum _j{B}_{ij}{\omega }_j\xi ,\hfill \\ & & d\xi =-\sum _i{C}_i{\omega }_{i}Y-\sum _{ij}{\omega }_i{B}_{ij}{Y}_j,\hfill \end{array}$$

where ${\omega }_{ij}$ is the connection form of the Möbius metric g and ${\omega }_{ij}+{\omega }_{ji}=0$. The tensors

$$A=\sum _{ij}{A}_{ij}{\omega }_i\otimes {\omega }_j,B=\sum _{ij}{B}_{ij}{\omega }_i\otimes {\omega }_j,C=\sum _i{C}_i{\omega }_i$$

are called the Blaschke tensor, the Möbius second fundamental form, and the Möbius form of f, respectively. The covariant derivatives of $C_i,A_{ij},B_{ij}$ are defined by

$$\begin{array}{ccc}& & \sum _j{C}_{i,j}{\omega }_j=d{C}_i+\sum _j{C}_j{\omega }_{ji},\hfill \\ & & \sum _k{A}_{ij,k}{\omega }_k=d{A}_{ij}+\sum _k{A}_{ik}{\omega }_{kj}+\sum _k{A}_{kj}{\omega }_{ki},\hfill \\ & & \sum _k{B}_{ij,k}{\omega }_k=d{B}_{ij}+\sum _k{B}_{ik}{\omega }_{kj}+\sum _k{B}_{kj}{\omega }_{ki}.\hfill \end{array}$$

The integrability conditions for the structure equations are given by

$$\begin{array}{ccc}& & A_{ij,k}-A_{ik,j}=B_{ik}{C}_j-B_{ij}{C}_k,\hfill \end{array}$$

(1)

$$\begin{array}{ccc}& & C_{i,j}-C_{j,i}=\sum _k(B_{ik}{A}_{kj}-B_{jk}{A}_{ki}),\hfill \end{array}$$

(2)

$$\begin{array}{ccc}& & B_{ij,k}-B_{ik,j}={\delta }_{ij}{C}_k-{\delta }_{ik}{C}_j,\hfill \end{array}$$

(3)

$$\begin{array}{ccc}& & R_{ijkl}=B_{ik}{B}_{jl}-B_{il}{B}_{jk}+{\delta }_{ik}{A}_{jl}+{\delta }_{jl}{A}_{ik}-{\delta }_{il}{A}_{jk}-{\delta }_{jk}{A}_{il},\hfill \end{array}$$

(4)

$$\begin{array}{ccc}& & R_{ij}:=\sum _k{R}_{ikjk}=-\sum _k{B}_{ik}{B}_{kj}+\left(tr\mathbf{A}\right){\delta }_{ij}+(n-2)A_{ij},\hfill \end{array}$$

(5)

$$\begin{array}{ccc}& & \sum _i{B}_{ii}=0,\sum _{ij}{\left(B_{ij}\right)}^2=\frac{n-1}{n},tr\mathbf{A}=\sum _i{A}_{ii}=\frac{1}{2n}(1+\frac{n}{n-1}R),\hfill \end{array}$$

(6)

Here, $R_{ijkl}$ denotes the curvature tensor of g, and $R={\sum }_{ij}{R}_{ijij}$ is the Möbius scalar curvature. We know that all coefficients in the structure equations are determined by $\{g,B\}$ when $n\ge 3$. Thus, we have the following:

Theorem 5

([21]). Two hypersurfaces $f:M^n\to {\mathbb{R}}^{n+1}$ and $\overline{f}:M^n\to {\mathbb{R}}^{n+1}(n\ge 3)$ are Möbius-equivalent if and only if there exists a diffeomorphism $\phi :M^n\to M^n$ which preserves the Möbius metric and the Möbius second fundamental form.

We would also like to recall from [20,21] how $\{B,C\}$ can be calculated in terms of the geometry of f in ${\mathbb{R}}^{n+1}$:

$$\begin{array}{cc}\hfill B_{ij}& ={\rho }^{-1}(h_{ij}-H{\delta }_{ij}),\hfill \\ \hfill C_i& =-{\rho }^{-2}[e_i\left(H\right)+\sum _j(h_{ij}-H{\delta }_{ij})e_j(log\rho )],\hfill \end{array}$$

(7)

where the Hessian and ∇ are with respect to $I=df·df$. The eigenvalues of B are called the Möbius principal curvatures of f. Let $\{b_1,\cdots ,b_n\}$ be the Möbius principal curvatures and $\{{\lambda }_1,\cdots ,{\lambda }_n\}$ be the principal curvatures of f. Then, from Equation (7), we have

$$b_i={\rho }^{-1}({\lambda }_i-H).$$

(8)

Clearly, the number of distinct Möbius principal curvatures is the same as that of the principal curvatures of f and

$${\mathbb{M}}_{ijk}=\frac{{\lambda }_i-{\lambda }_j}{{\lambda }_i-{\lambda }_k}=\frac{b_i-b_j}{b_i-b_k},$$

(9)

which confirms that the Möbius curvatures are Möbius invariants.

3. Some Local Properties of the Generic Conformally Flat Hypersurfaces

In this section, we recall some facts about the generic conformally flat hypersurfaces and prove Theorem 1 and Proposition 1.

Let $(M^m,g),m\ge 3$ be a Riemannian manifold and ∇ be its Riemannian connection. The Riemannian curvature $(0,4)$ tensor is defined as follows:

$$R_m(X,Y,Z,W)=g({\nabla }_Y{\nabla }_{X}Z-{\nabla }_X{\nabla }_{Y}Z+{\nabla }_{[X,Y]}Z,W),X,Y,Z,W\in \mathsf{\Gamma }\left(T{M}^m\right).$$

If the Riemannian curvature tensor vanishes (i.e., $R_m\equiv 0$), then the Riemannian metric g is flat.

Let $\{e_1,\cdots ,e_m\}$ be a local orthonormal basis for $T{M}^m$ with a dual basis $\{{\omega }_1,\cdots ,{\omega }_m\}$. The Ricci curvature $Ric$ and scalar curvature R are defined by

$$Ric(X,Y)=\sum _{i=1}^m{R}_m(X,e_i,Y,e_i),R=\sum _{i=1}^{m}Ric(e_i,e_i).$$

The Schouten tensor S is the symmetric $(0,2)$ tensor, defined by

$$S=\frac{1}{m-2}\left(Ric-\frac{R}{2(m-1)}g\right).$$

Let $R_{ijkl}=R_m(e_i,e_j,e_k,e_l),Ri{c}_{ij}=Ric(e_i,e_j)$, and $S_{ij}=S(e_i,e_j)$. Then, we have

$$R_m=\sum _{ijkl}{R}_{ijkl}{\omega }_i\otimes {\omega }_j\otimes {\omega }_k\otimes {\omega }_l,Ric=\sum _{ij}Ri{c}_{ij}{\omega }_i\otimes {\omega }_j,S=\sum _{ij}{S}_{ij}{\omega }_i\otimes {\omega }_j.$$

The Weyl tensor W is the $(0,4)$ tensor defined by

$$\begin{array}{cc}\hfill W_{ijkl}=& R_{ijkl}-\frac{1}{m-2}\left(Ri{c}_{ik}{\delta }_{jl}+Ri{c}_{jl}{\delta }_{ik}-Ri{c}_{jk}{\delta }_{il}-Ri{c}_{il}{\delta }_{jk}\right)\hfill \\ \hfill & +\frac{R}{(m-1)(m-2)}({\delta }_{ik}{\delta }_{jl}-{\delta }_{il}{\delta }_{jk}).\hfill \end{array}$$

With the Schouten tensor, the Weyl tensor also can be defined by

$$W_{ijkl}=R_{ijkl}-\left(S_{ik}{\delta }_{jl}+S_{jl}{\delta }_{ik}-S_{jk}{\delta }_{il}-S_{il}{\delta }_{jk}\right).$$

When the dimension $m=3$, the Weyl tensor $W\equiv 0$. The Weyl tensor is an algebraic curvature which is traceless.

Let $S_{ij,k}$ be the coefficients of the covariant derivative of S. Then, the Cotten tensor T is defined by

$$T_{ijk}=S_{ij,k}-S_{ik,j}.$$

Definition 1.

Two Riemannian metrics g and $\overline{g}$ on the manifold $M^m$ are conformal if there is a smooth function ρ on $M^m$ such that $\overline{g}=e^{2\rho }g.$

Let $\overline{W}$ be the Weyl tensor of the metric $\overline{g}=e^{2\rho }g$. Then, we have

$${\overline{W}}_{ijkl}=e^{2\rho }{W}_{ijkl}.$$

When the dimension $m=3$, let $\overline{T}$ be the Cotten tensor of the metric $\overline{g}=e^{2\rho }g$. Then, we have

$${\overline{T}}_{ijk}=T_{ijk}.$$

Definition 2.

The Riemannian manifold $(M^m,g)$ is said to be conformally flat if, for any $p\in M^m$, there is an open set $U\subset M^m$ containing p and a smooth function $\rho \in C^{\infty }\left(U\right)$ such that the Riemannian metric $e^{2\rho }g$ is flat.

The following results are due to Weyl’s work:

Theorem 6.

(i) Any two-dimensional Riemannian manifold is conformally flat. (ii) A three-dimensional Riemannian manifold is conformally flat if and only if the Cotten tensor $T\equiv 0$. (iii) If the dimensions $m\ge 4$, then the Riemannian manifold $(M^m,g)$ is conformally flat if and only if the Weyl tensor $W\equiv 0$.

Let $f:M^3\to {\mathbb{R}}^4$ be a generic hypersurface. Since f is without an umbilical point, the Möbius metric g, the Möbius second fundamental form B, the Blaschke tensor A, and the Möbius form C can be defined. We chose a local orthonormal basis $\{E_1,E_2,E_3\}$ with respect to the Möbius metric g such that

$$\left(B_{ij}\right)=diag\{b_1,b_2,b_3\},b_1<b_2<b_3.$$

(10)

Let $\{{\omega }_1,{\omega }_2,{\omega }_3\}$ be the dual basis of $\{E_1,E_2,E_3\}$. The conformal fundamental forms of f are defined by

$${\mathsf{\Theta }}_1=\sqrt{(b_3-b_1)(b_2-b_1)}{\omega }_1,{\mathsf{\Theta }}_2=\sqrt{(b_3-b_2)(b_2-b_1)}{\omega }_2,{\mathsf{\Theta }}_3=\sqrt{(b_3-b_1)(b_3-b_2)}{\omega }_3.$$

Using Equations (5) and (6), the Schouten tensor of f with respect to the Möbius metric g is

$$S=\sum _{ij}{S}_{ij}{\omega }_i\otimes {\omega }_j=\sum _{ij}(-\sum _k{B}_{ik}{B}_{kj}+A_{ij}+\frac{1}{6}{\delta }_{ij}){\omega }_i\otimes {\omega }_j.$$

Thus, we have

$$S_{ij,k}=-\sum _l(B_{il,k}{B}_{lj}+B_{il}{B}_{lj,k})+A_{ij,k}.$$

(11)

If the hypersurface f is conformally flat, then $S_{ij,k}=S_{ik,j}$. By combining Equations (1) and (10), we obtain the following equation:

$$b_k{B}_{ik,j}-b_j{B}_{ij,k}=2(B_{ij}{C}_k-B_{ik}{C}_j).$$

(12)

Using Equation (3), we have the following equations:

$$\begin{array}{cc}\hfill & B_{12,3}=B_{13,2}=0,\hfill \\ \hfill & B_{ij,i}=\frac{3b_i}{b_j-b_i}{C}_j,B_{ii,j}=\frac{b_i-b_k}{b_j-b_i}{C}_j,i\ne j,j\ne k,i\ne k.\hfill \end{array}$$

(13)

Using $d{B}_{ij}+{\sum }_k{B}_{kj}{\omega }_{ki}+{\sum }_k{B}_{ik}{\omega }_{kj}={\sum }_k{B}_{ij,k}{\omega }_k$ and Equation (10), we obtain

$${\omega }_{ij}=\sum _k\frac{B_{ij,k}}{b_i-b_j}{\omega }_k=\frac{B_{ij,i}}{b_i-b_j}{\omega }_i+\frac{B_{ij,j}}{b_i-b_j}{\omega }_j,i\ne j,1\le i,j\le 3.$$

(14)

The following lemma is trivial under Equations (13) and (14) (alternatively, see [8,12]):

Lemma 1

([8,12]). Let $M^3\to {\mathbb{R}}^4$ be a generic hypersurface. Then, the following are equivalent:

(1) The hypersurface is conformally flat;

(2) The Schouten tensor is a Codazzi tensor;

(3) The conformal fundamental forms ${\mathsf{\Theta }}_1,{\mathsf{\Theta }}_2$, and ${\mathsf{\Theta }}_3$ are closed.

Next, we give the standard examples of generic conformally flat hypersurfaces in ${\mathbb{R}}^4$:

Example 1.

Let $u:M^2⟶{\mathbb{R}}^3$ be an immersed surface. We define the cylinder over u in ${\mathbb{R}}^4$ as

$$f=(id,u):{\mathbb{R}}^1\times M^2⟶{\mathbb{R}}^1\times {\mathbb{R}}^3={\mathbb{R}}^4,f(t,y)=(t,u\left(x\right)),$$

where $id:{\mathbb{R}}^1⟶{\mathbb{R}}^1$ is the identity map. If the surface u is of a constant Gaussian curvature, then the cylinder over u is conformally flat.

Example 2.

Let $u:M^2⟶{\mathbb{S}}^3\subset {\mathbb{R}}^4$ be an immersed surface. We define the cone over u in ${\mathbb{R}}^4$ as

$$f:{\mathbb{R}}^{+}\times M^2⟶{\mathbb{R}}^4,f(t,x)=tu\left(x\right).$$

If the surface u is of a constant Gaussian curvature, then the cone over u is conformally flat.

Let $x_1:{\mathbb{S}}^1\left(r\right)\to {\mathbb{R}}^2$ be the circle with a radius r and $x_2:{\mathbb{S}}^1\left(\sqrt{1-r^2}\right)\to {\mathbb{R}}^2$ be the circle with a radius $\sqrt{1-r^2}$. Then, the homogeneous torus $x:{\mathbb{S}}^1\left(r\right)\times {\mathbb{S}}^1\left(\sqrt{1-r^2}\right)\to {\mathbb{S}}^3$ is defined as follows:

$$x=(x_1,x_2):{\mathbb{S}}^1\left(r\right)\times {\mathbb{S}}^1\left(\sqrt{1-r^2}\right)\to {\mathbb{S}}^3\subset {\mathbb{R}}^2\times {\mathbb{R}}^2={\mathbb{R}}^4.$$

The homogeneous torus x is a flat surface in ${\mathbb{S}}^3$, and thus the cone over the homogeneous torus is conformally flat.

Example 3.

Let ${\mathbb{R}}_{+}^3=\{(x_1,x_2,x_3)\in {\mathbb{R}}^3|x_3>0\}$ be the upper half-space endowed with the standard hyperbolic metric

$$d{s}^2=\frac{1}{x_3^2}[d{x}_1^2+d{x}_2^2+d{x}_3^2].$$

Let $u=(x_1,x_2,x_3):M^2⟶{\mathbb{R}}_{+}^3$ be an immersed surface. We define the rotational hypersurface over u in ${\mathbb{R}}^4$ as

$$f:{\mathbb{S}}^1\times M^2⟶{\mathbb{R}}^4,f(\varphi ,x_1,x_2,x_3)=(x_1,x_2,x_3\varphi ),$$

where $\varphi :{\mathbb{S}}^1⟶{\mathbb{S}}^1$ is the unit circle. If the surface u is of a constant Gaussian curvature, then the rotational hypersurface over u is conformally flat.

Before the proof of Theorem 1, we need the following results, given in [24]:

Theorem 7

([24]). Let $f:M^3\to {\mathbb{R}}^4$ be an immersed hypersurface with three distinct principal curvatures. If f is of a constant Möbius sectional curvature c, then f is Möbius equivalent to a cone over a flat torus $x:{\mathbb{S}}^1\left(r\right)\times {\mathbb{S}}^1\left(\sqrt{1-r^2}\right)\to {\mathbb{S}}^3$ with a Möbius sectional curvature $c=0$.

Proof. 

Next, we prove Theorem 1. Since the Möbius form vanishes, $C=0$. Then, under Equation (13), we have

$$B_{12,3}=0,B_{ii,j}=B_{ij,i}=0,i\ne j.$$

Since ${\sum }_i{B}_{ii}=0$, then ${\sum }_i{B}_{ii,j}=0,1\le j\le 3.$ Thus, we obtain

$$B_{ij,k}=0,1\le i,j,k\le 3.$$

According to Equation (14), we have ${\omega }_{ij}=0,1\le i,j,\le 3$, which implies that the hypersurface is flat. Using Theorem 7, we finish the proof of Theorem 1. □

Proof. 

Next, we prove Proposition 1. Let $f:M^3\to {\mathbb{R}}^4$ be a generic conformally flat hypersurface with the Möbius curvature $M_{123}$ being constant. Let $\{b_1,b_2,b_3\}$ be the Möbius principal curvatures and $M_{123}=\frac{b_1-b_2}{b_1-b_3}.$ Thus, combined with Equation (6), we have

$$b_1+b_2+b_3=0,b_1^2+b_2^2+b_3^2=\frac{2}{3},b_1-b_2=M_{123}(b_1-b_3),$$

which implies that the Möbius principal curvatures $\{b_1,b_2,b_3\}$ are constant. From Equation (13), we find that $C=0$, and with Theorem 1, we finish the proof of Proposition 1. □

4. Some Global Behavior of the Compact, Generic Conformally Flat Hypersurfaces

In this section, we investigate the global behavior of the compact, generic conformally flat hypersurfaces and give the proof of Theorems 2 and 3 as well as Corollaries 2 and 1.

Let $f:M^3\to {\mathbb{R}}^4$ be a generic conformally flat hypersurface. Let $A={\sum }_{ij}{A}_{ij}{\omega }_i\otimes {\omega }_j,B={\sum }_{ij}{B}_{ij}{\omega }_i\otimes {\omega }_j$, and $C={\sum }_i{C}_i{\omega }_i$ denote the Blaschke tensor, the Möbius second fundamental form, and the Möbius form, respectively. We say that the pair $(U,\omega )$ is admissible if the following are true:

(1)

U is an open subset of $M^3$;

(2)

$\omega =({\omega }_1,{\omega }_2,{\omega }_3)$ is an orthonormal co-frame field on U with respect to the Möbius metric g;

(3)

${\omega }_1\wedge {\omega }_2\wedge {\omega }_3=d{v}_g$;

(4)

$B={\sum }_i{b}_i{\omega }_i\otimes {\omega }_i$.

Denote with $F=(E_1,E_2,E_3)$ the dual frame field of $\omega$. Then, it is easily seen that $(U,\omega )$ is admissible if and only if $E_i$ is a unit principal vector associated with $b_i$ for each $1\le i\le 3$, and $\{E_1,E_2,E_3\}$ is an oriented basis associated with the orientation of $M^3$. Denote with $\left\{{\omega }_{ij}\right\}$ the connection form with respect to $(U,{\omega }_1,{\omega }_2,{\omega }_3)$. Thus, under the admissible frame field $\{E_1,E_2,E_3\}$, we have

$$\left(B_{ij}\right)=diag\{b_1,b_2,b_3\}.$$

Now, we introduce two two-forms $\mathsf{\Phi }$ and $\mathsf{\Psi }$ on $M^3$. For every admissible co-frame field $(U,\omega )$, we set

$$\begin{array}{cc}\hfill & \mathsf{\Phi }={\omega }_{12}\wedge {\omega }_3+{\omega }_{23}\wedge {\omega }_1+{\omega }_{31}\wedge {\omega }_2,\hfill \\ \hfill & \mathsf{\Psi }={(b_1-b_2)}^2{\omega }_{12}\wedge {\omega }_3+{(b_2-b_3)}^2{\omega }_{23}\wedge {\omega }_1+{(b_1-b_3)}^2{\omega }_{31}\wedge {\omega }_2.\hfill \end{array}$$

If $(U,\omega )$ and $(\tilde{U},\tilde{\omega })$ are both admissible co-frame fields with $U\cap \tilde{U}\ne \varnothing$, then on $U\cap \tilde{U}$, ${\omega }_i={ϵ}_i{\tilde{\omega }}_i,{\omega }_{ij}={ϵ}_i{ϵ}_j{\tilde{\omega }}_{ij}$ for every $1\le i\le 3$, where ${ϵ}_i=1$ or $-1$ and ${ϵ}_1{ϵ}_2{ϵ}_3=1$. Thus, we have

$${\omega }_{12}\wedge {\omega }_3={\tilde{\omega }}_{12}\wedge {\tilde{\omega }}_3,{\omega }_{23}\wedge {\omega }_1={\tilde{\omega }}_{23}\wedge {\tilde{\omega }}_1,{\omega }_{31}\wedge {\omega }_2={\tilde{\omega }}_{31}\wedge {\tilde{\omega }}_2.$$

Therefore, the two forms $\mathsf{\Phi }$ and $\mathsf{\Psi }$ are well-defined on $M^3$.

By combining $d{\omega }_{ij}-{\sum }_k{\omega }_{ik}\wedge {\omega }_{kj}=\frac{-1}{2}{\sum }_{kl}{R}_{ijkl}{\omega }_k\wedge {\omega }_l$, $d{\omega }_i={\sum }_k{\omega }_{ik}\wedge {\omega }_k$, and Equations (13) and (14), we obtain

$$\begin{array}{cc}\hfill & d({\omega }_{12}\wedge {\omega }_3)=-R_{1212}{\omega }_1\wedge {\omega }_2\wedge {\omega }_3\hfill \\ \hfill & +[\frac{-9b_1{b}_2{C}_3^2}{{(b_1-b_3)}^2{(b_2-b_3)}^2}+\frac{9b_2{b}_3{C}_1^2}{{(b_1-b_2)}^2{(b_1-b_3)}^2}+\frac{9b_1{b}_3{C}_2^2}{{(b_1-b_2)}^2{(b_2-b_3)}^2}]{\omega }_1\wedge {\omega }_2\wedge {\omega }_3.\hfill \end{array}$$

Similarly, we can compute $d({\omega }_{23}\wedge {\omega }_1)$ and $d({\omega }_{31}\wedge {\omega }_2)$. Therefore, we have

$$\begin{array}{cc}\hfill & d\mathsf{\Phi }=[\frac{9b_1{b}_2{C}_3^2}{{(b_1-b_3)}^2{(b_2-b_3)}^2}-R_{1212}-R_{1313}-R_{2323}]{\omega }_1\wedge {\omega }_2\wedge {\omega }_3\hfill \\ \hfill & +[\frac{9b_2{b}_3{C}_1^2}{{(b_1-b_2)}^2{(b_1-b_3)}^2}+\frac{9b_1{b}_3{C}_2^2}{{(b_1-b_2)}^2{(b_2-b_3)}^2}]{\omega }_1\wedge {\omega }_2\wedge {\omega }_3.\hfill \end{array}$$

(15)

By using $d{b}_i={\sum }_k{B}_{ii,k}{\omega }_k$ and the same computation as $d\mathsf{\Phi }$, we can obtain

$$\begin{array}{cc}\hfill & d\mathsf{\Psi }=-[{(b_1-b_2)}^2{R}_{1212}+{(b_1-b_3)}^2{R}_{1313}+{(b_2-b_3)}^2{R}_{2323}]d{v}_g\hfill \\ \hfill & +[\frac{18b_1{b}_2{C}_3^2}{{(b_1-b_3)}^2{(b_2-b_3)}^2}+\frac{18b_2{b}_3{C}_1^2}{{(b_1-b_2)}^2{(b_1-b_3)}^2}+\frac{18b_1{b}_3{C}_2^2}{{(b_1-b_2)}^2{(b_2-b_3)}^2}]d{v}_g,\hfill \end{array}$$

(16)

where $d{v}_g={\omega }_1\wedge {\omega }_2\wedge {\omega }_3$. By combining Equation (15) and Equation (16), we have

$$2d\mathsf{\Phi }-d\mathsf{\Psi }=\{[{(b_1-b_2)}^2-2]R_{1212}+[{(b_2-b_3)}^2-2]R_{2323}+[{(b_1-b_3)}^2-2]R_{1313}\}d{v}_g.$$

(17)

If $M^3$ is compact, then Equation (17) implies that

$${\int }_{M^3}\{[{(b_1-b_2)}^2-2]R_{1212}+[{(b_2-b_3)}^2-2]R_{2323}+[{(b_1-b_3)}^2-2]R_{1313}\}d{v}_g=0.$$

(18)

Let $R_1=R_{1212}+R_{1313},R_2=R_{1212}+R_{2323}$, and $R_3=R_{1313}+R_{2323}$ be the Ricci curvatures. Then, we have

$$R_{1212}=\frac{1}{2}(R_1+R_2-R_3),R_{1313}=\frac{1}{2}(R_1+R_3-R_2),R_{2323}=\frac{1}{2}(R_2+R_3-R_1).$$

By combining $b_1+b_2+b_3=0$ and $b_1^2+b_2^2+b_3^2=\frac{2}{3}$, we can derive that

$$\begin{array}{cc}\hfill & [{(b_1-b_2)}^2-2]R_{1212}+[{(b_2-b_3)}^2-2]R_{2323}+[{(b_1-b_3)}^2-2]R_{1313}\hfill \\ \hfill & =3\left[R_1(b_1^2-\frac{4}{9})+R_2(b_2^2-\frac{4}{9})+R_3(b_3^2-\frac{4}{9})\right].\hfill \end{array}$$

Thus from Equation (18), we have

$${\int }_{M^3}\left[R_1(b_1^2-\frac{4}{9})+R_2(b_2^2-\frac{4}{9})+R_3(b_3^2-\frac{4}{9})\right]d{v}_g=0.$$

(19)

From $b_1+b_2+b_3=0$ and $b_1^2+b_2^2+b_3^2=\frac{2}{3}$, we find ${(b_1-b_2)}^2=\frac{4}{3}-3b_3^2>0$, and thus we have

$$b_1^2<\frac{4}{9},b_2^2<\frac{4}{9},b_3^2<\frac{4}{9}.$$

(20)

Proof. 

Now, we will prove Theorem 2. If the Ricci curvature of the Möbius metric is nonpositive pointwise ( or nonnegative pointwise), then $R_1\le 0,R_2\le 0$, and $R_3\le 0$ (or $R_1\ge 0,R_2\ge 0$, and $R_3\ge 0$). Thus, we have

$$R_1(b_1^2-\frac{4}{9})\le 0,R_2(b_2^2-\frac{4}{9})\le 0,R_3(b_3^2-\frac{4}{9})\le 0.$$

From Equation (19), we find that $R_1=R_2=R_3=0$, and thus the Möbius sectional curvatures $R_{1212},R_{1313}$, and $R_{2323}$ are zero. Therefore, the hypersurface is noncompact under Theorem 7, which is a contradiction with the statement that the hypersurface is compact. Theorem 2 is proven. □

Proof. 

From $b_1+b_2+b_3=0$ and $b_1^2+b_2^2+b_3^2=\frac{2}{3}$, we have $b_1^4+b_2^4+b_3^4=\frac{2}{9}$, and thus

$$\begin{array}{cc}\hfill & R_1(b_1^2-\frac{4}{9})+R_2(b_2^2-\frac{4}{9})+R_3(b_3^2-\frac{4}{9})\hfill \\ \hfill & =R_1{b}_1^2+R_2{b}_2^2+R_3{b}_3^2-\frac{4}{9}R\hfill \\ \hfill & \le \frac{\sqrt{2}}{3}\left|Ric\right|-\frac{4}{9}R=\frac{\sqrt{2}}{3}\left[\left|Ric\right|-\frac{2\sqrt{2}}{3}R\right]\hfill \end{array}$$

If the equality holds pointwise on the global hypersurface f in the above inequality, then $R_1=\lambda b_1^2,R_2=\lambda b_2^2$, and $R_3=\lambda b_3^2$ on f, which implies that the Ricci curvature has a sign. Under Theorem 2, we know this is impossible. Through combination with Equation (19), Corollary 1 is proved. □

Proof. 

Using Equations (4) and (6), we have

$$\begin{array}{cc}\hfill & [{(b_1-b_2)}^2-2]R_{1212}+[{(b_2-b_3)}^2-2]R_{2323}+[{(b_1-b_3)}^2-2]R_{1313}\hfill \\ \hfill & =\frac{2}{9}-\frac{10}{3}tr\left(A\right)+3[b_1^2{a}_1+b_2^2{a}_2+b_3^2{a}_3].\hfill \end{array}$$

(21)

On the other hand, Equation (5) implies that

$${\left|Ric\right|}^2=\frac{2}{9}+5tr{\left(A\right)}^2+{\left|A\right|}^2-\frac{4}{3}tr\left(A\right)-2[b_1^2{a}_1+b_2^2{a}_2+b_3^2{a}_3],$$

(22)

where $\left|Ric\right|$ denotes the norm of the Ricci curvature. By combining Equations (21) and (22), we can derive that

$$\begin{array}{cc}\hfill & [{(b_1-b_2)}^2-2]R_{1212}+[{(b_2-b_3)}^2-2]R_{2323}+[{(b_1-b_3)}^2-2]R_{1313}\hfill \\ \hfill & =\frac{5}{9}-\frac{16}{3}tr\left(A\right)+\frac{15}{2}tr{\left(A\right)}^2+\frac{3}{2}{\left|A\right|}^2-\frac{3}{2}{\left|Ric\right|}^2.\hfill \end{array}$$

(23)

Let $\tilde{A}:=A-\frac{1}{3}tr\left(A\right)g$ denote the trace-free Blaschke tensor. Then, $|\tilde{A}{|}^2={\left|A\right|}^2-\frac{1}{3}tr{\left(A\right)}^2$. Thus, from Equation (23), we have

$$\begin{array}{cc}\hfill & [{(b_1-b_2)}^2-2]R_{1212}+[{(b_2-b_3)}^2-2]R_{2323}+[{(b_1-b_3)}^2-2]R_{1313}\hfill \\ \hfill & =\frac{3}{2}|\tilde{A}{|}^2-\frac{3}{2}{\left[\right|Ric|}^2-\frac{1}{3}{R}^2]-\frac{2}{3}R-\frac{1}{9}.\hfill \end{array}$$

(24)

Now, if the hypersurface $M^3$ is compact, then

$${\int }_{M^3}\left[|\tilde{A}{|}^2+\frac{1}{3}{R}^2-\frac{4}{9}R-{\left|Ric\right|}^2-\frac{2}{27}\right]d{v}_g=0.$$

(25)

Therefore, we finish the proof of Theorem 3. □

Proof. 

Since $R=tr\left(Ric\right)$, we have ${\left|Ric\right|}^2-\frac{1}{3}{R}^2\ge 0$ on $M^3$. If ${\left|Ric\right|}^2-\frac{1}{3}{R}^2\equiv 0$, then $Ric=\frac{1}{3}Rg$, and the hypersurface is an Einstein type. Since the dimension of the hypersurface is three, the sectional curvatures are constant. From the results in [24,25], we know that the hypersurface is locally Möbius equivalent to a cone over a homogeneous torus $x:{\mathbb{S}}^1\left(r\right)\times {\mathbb{S}}^1\left(\sqrt{1-r^2}\right)\to {\mathbb{S}}^3\subset {\mathbb{R}}^4$ in three-sphere ${\mathbb{S}}^3$, which is a contradiction with the idea that the hypersurface is compact. Thus, ${\left|Ric\right|}^2-\frac{1}{3}{R}^2\ne 0$ and

$${\int }_{M^3}\left[{\left|Ric\right|}^2-\frac{1}{3}{R}^2\right]d{v}_g>0.$$

Since $R=4tr\left(A\right)-\frac{2}{3}$ under Equation (6), by combining this with Equation (25), we have

$${\int }_{M^3}\left[|\tilde{A}{|}^2-\frac{16}{9}tr\left(A\right)+\frac{2}{9}\right]d{v}_g={\int }_{M^3}\left[{\left|Ric\right|}^2-\frac{1}{3}{R}^2\right]d{v}_g>0.$$

Thus, Corollary 2 is proven. □