## Hypersurfaces with isotropic Blaschke tensor.(English)Zbl 1242.53010

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 $$3$$.

### MSC:

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

Zbl 0912.53012
Full Text:

### References:

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