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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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

Permanent link to this document
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
Secondary: 53B25: Local submanifolds [See also 53C40]

Keywords
Möbius metric Möbius sectional curvature Möbius Ricci curvature

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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