Hypersurfaces with isotropic Blaschke tensor. (English) Zbl 1242.53010

Guo, Zhen; Fang, Jianbo; Lin, Limiao

J. Math. Soc. Japan 63, No. 4, 1155-1186 (2011).

For a connected smooth \(n\)-dimensional hypersurface \(x: M^n \rightarrow \mathbb S^{n+1}\) without umbilical points, there exist three basic Möbius invariants, namely the Möbius metric \(g\), the Möbius form, the Blaschke tensor \(A\). We say that \(M\) is a hypersurface with isotropic Blaschke tensor if and only if there exists a function \(\lambda\) such that \(A= \lambda g\). Note that in this case this corresponds to the classical definition of an isotropic tensor in a Riemannian manifold, following O’Neill, saying that the length of \(A(v,\dots,v)\) is independent of the unit vector \(v\).
The hypersurface is called constant isotropic if \(\lambda\) is a constant function on \(M\). In this case a classification was obtained in [C. Wang, Manuscr. Math. 96, No. 4, 517–534 (1998; Zbl 0912.53012)].
Here the authors obtain a complete classification, up to Möbius equivalence, when \(\lambda\) is non constant, provided the dimension is at least \(3\).

Reviewer: Luc Vrancken (Valenciennes)

MSC:

53A30	Conformal differential geometry (MSC2010)
53B25	Local submanifolds

Keywords:

Möbius invariants; Möbius metric; Möbius form; Blaschke tensor; Möbius second fundamental form

Citations:

Zbl 0912.53012

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