## Tohoku Mathematical Journal

### Classification of hypersurfaces with constant Möbius Ricci curvature in $\mathbb{R}^{n+1}$

#### Abstract

Let $f: M^n\rightarrow \mathbb{R}^{n+1}$ be an immersed umbilic-free hypersurface in an $(n+1)$-dimensional Euclidean space $\mathbb{R}^{n+1}$ with standard metric $I=df\cdot df$. Let $II$ be the second fundamental form of the hypersurface $f$. One can define the Möbius metric $g=\frac{n}{n-1}(\|II\|^2-n|{\rm tr}II|^2)I$ on $f$ which is invariant under the conformal transformations (or the Möbius transformations) of $\mathbb{R}^{n+1}$. The sectional curvature, Ricci curvature with respect to the Möbius metric $g$ is called Möbius sectional curvature, Möbius Ricci curvature, respectively. The purpose of this paper is to classify hypersurfaces with constant Möbius Ricci curvature.

#### Article information

Source
Tohoku Math. J. (2), Volume 67, Number 3 (2015), 383-403.

Dates
First available in Project Euclid: 6 November 2015

https://projecteuclid.org/euclid.tmj/1446818558

Digital Object Identifier
doi:10.2748/tmj/1446818558

Mathematical Reviews number (MathSciNet)
MR3420551

Zentralblatt MATH identifier
1333.53019

Subjects
Primary: 53A30: Conformal differential geometry

#### Citation

Guo, Zhen; Li, Tongzhu; Wang, Changping. Classification of hypersurfaces with constant Möbius Ricci curvature in $\mathbb{R}^{n+1}$. Tohoku Math. J. (2) 67 (2015), no. 3, 383--403. doi:10.2748/tmj/1446818558. https://projecteuclid.org/euclid.tmj/1446818558

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