On projectively equivalent metrics near points of bifurcation

Details On projectively equivalent metrics near points of bifurcation On projectively equivalent metrics near points of bifurcation

2008-09-21

arxiv.org/abs/0809.3602v1

Topologival methods in the theory of integrable systems (Eds.: Bolsinov A.V., Fomenko A.T., Oshemkov A.A.), Cambridge scientif

Mathematics

Differential Geometry

28 pages; no figures

Scientific paper

Let Riemannian metrics $g$ and $\bar g$ on a connected manifold $M^n$ have the same geodesics (considered as unparameterized curves). Suppose the eigenvalues of one metric with respect to the other are all different at a point. Then, by the famous Levi-Civita's Theorem, the metrics have a certain standard form near the point. Our main result is a generalization of Levi-Civita's Theorem for the points where the eigenvalues of one metric with respect to the other bifurcate.

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