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Geometric construction of the Levi-Civita parallelism. (English) Zbl 0923.51016
This paper is devoted to a construction of the Levi-Civita connection associated to a pseudo-Riemannian metric on a manifold in terms of synthetic differential geometry.
The author first reviews some background material on affine connections in synthetic differential geometry and proves an interpretation of symmetric affine connections in terms of infinitesimal 2-simplices. Next, pseudo-Riemannian metrics are introduced as quadratic differential forms which satisfy a certain non-degeneracy condition. It is proved, that any quadratic differential form has a unique symmetric extension to a cubic differential form, and using this, the notion of geodesic midpoint is introduced in presence of a pseudo-Riemannian metric. It is shown, that this midpoint can be given a variational characterization. Using the geodesic midpoints the author then associates a symmetric affine connection to any pseudo-Riemannian metric and proves that this equals the Levi-Civita connection by computing Christoffel symbols.
Reviewer: A.Cap (Wien)

51K10 Synthetic differential geometry
53B20 Local Riemannian geometry
58A03 Topos-theoretic approach to differentiable manifolds
18F99 Categories in geometry and topology
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