Tohoku Mathematical Journal
2015
September
SECOND SERIES VOL. 67, NO. 3

Tohoku Math. J.
67 (2015), 383-403

Title CLASSIFICATION OF HYPERSURFACES WITH CONSTANT MÖBIUS RICCI CURVATURE IN $\mathbb{R}^{N+1}$

Author Zhen Guo, Tongzhu Li and Changping Wang

(Received September 10, 2013, revised May 26, 2014)
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.

Mathematics Subject Classification. Primary 53A30; Secondary 53B25.
Key words and phrases. Möbius metric, Möbius sectional curvature, Möbius Ricci curvature.

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