Tohoku Mathematical Journal

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

Zhen Guo, Tongzhu Li, and Changping Wang

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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

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

First available in Project Euclid: 6 November 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53A30: Conformal differential geometry
Secondary: 53B25: Local submanifolds [See also 53C40]

Möbius metric Möbius sectional curvature Möbius Ricci curvature


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.

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