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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
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