On natural connections on Riemannian manifolds.(English)Zbl 0679.53025

D. B. A. Epstein [J. Differ. Geom. 10, 631-645 (1975; Zbl 0321.53039)] calls a natural connection a rule which assigns to each Riemannian manifold $$(M,g)$$ a connection $$\Gamma_{(M,g)}$$ satisfying $$f*\Gamma_{(M,g)}=\Gamma_{(M',g')}$$ for any local isometry $$f:(M,g)\to (M',g')$$. He proves that the Levi-Cività connection is the only first order natural connection under some polynomiality assumptions.
The present paper gives a simple proof of this uniqueness of the Levi- Cività connection without the additional polynomiality assumptions. In addition, the following variant is proved: The Levi-Cività connection is the only first order natural connection with respect to orientation preserving local diffeomorphisms in dimensions not equal one or three. All natural connections on 3-dimensional pseudo-Riemannian manifolds of signature s form a one-dimensional parameter family $$\Gamma =\Gamma_{\text{Levi-Cività}}+cV$$, where $$c\in \mathbb{R}$$ and $$V$$ is the zero order operator induced by the vector product. In dimension one, there is a similar one-parameter family, with $$V$$ being now the zero order operator induced by the scalar product.
Reviewer: J.Virsik

MSC:

 53C05 Connections (general theory) 53A55 Differential invariants (local theory), geometric objects 58A20 Jets in global analysis

Zbl 0321.53039
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