arXiv subjects

Vladimir S. Matveev

Publications and source records attributed to Vladimir S. Matveev.

100 records · Page 6Linked to original sources

arXiv2008

Metric connections in projective differential geometry

We search for Riemannian metrics whose Levi-Civita connection belongs to a given projective class. Following Sinjukov and Mikes, we show that such metrics correspond precisely to suitably positive solutions of a certain projectively invariant finite-type linear system of partial differential equations. Prolonging this system, we may reformulate these equations as defining covariant constant sections of a certain vector bundle with connection. This vector bundle and its connection are derived from the Cartan connection of the underlying projective structure.

math.DG↗

arXiv2008

Fubini Theorem for pseudo-Riemannian metrics

We generalize the following classical result of Fubini for pseudo-Riemannian metrics: if three essentially different metrics on $M^{n\ge 3}$ share the same unparametrized geodesics, and two of them (say, $g$ and $\bar g$) are strictly nonproportional (i.e., the minimal polynomial of $g^{iα} \bar g_{αj}$ coincides with the characteristic polynomial) at least at one point, then they have constant curvature.

math.DG↗

arXiv2008

Appendix: Dini theorem for pseudo-Riemannian metrics

We construct local normal forms of pseudo-Riemannian projectively equivalent 2-dimensional metrics.

math.DG↗

arXiv2007

A solution of a problem of Sophus Lie: Normal forms of 2-dim metrics admitting two projective vector fields

We give a complete list of normal forms for the 2-dimensional metrics that admit a transitive Lie pseudogroup of geodesic-preserving transformations and we show that these normal forms are mutually non-isometric. This solves a problem posed by Sophus Lie.

math.DG↗

arXiv2004

Strictly non-proportional geodesically equivalent metrics have $h_\text{top}(g)=0$

Suppose the Riemannian metrics $g$ and $\bar g$ on a closed connected manifold $M^n$ are geodesically equivalent and strictly non-proportional at least at one point. Then the topological entropy of the geodesic flow of $g$ vanishes.

math.DG↗

arXiv2004

Projective Lichnerowicz-Obata conjecture

We solve two classical conjectures by showing that if an action of a connected Lie group on a complete Riemannian manifold preserves the geodesics (considered as unparameterized curves), then the metric has constant positive sectional curvature, or the group acts by affine transformations.

math.DG↗

arXiv2004

Closed manifolds admitting metrics with the same geodesics

The goal of this survey is to give a list of resent results about topology of manifolds admitting different metrics with the same geodesics. We emphasize the role of the theory of integrable systems in obtaining these results.

math.DG↗

arXiv2004

A new integrable system on the sphere

We present a new Liouville-integrable natural Hamiltonian system on the (cotangent bundle of the) two-dimensional sphere. The second integral is cubic in the momenta.

math.DS↗

arXiv1999

Geodesic equivalence and integrability

We suggest a construction that, given a trajectorial diffeomorphism between two Hamiltonian systems, produces integrals of them. As the main example we treat geodesic equivalence of metrics. We show that the existence of a non-trivially geodesically equivalent metric leads to Liouville integrability, and present explicit formulae for integrals.

math.DG↗

arXiv1999

Dynamical and Topological methods in Theory of Geodesically Equivalent Metrics

This paper is a review on recently found connection between geodesically equivalent metrics and integrable geodesic flows. Suppose two different metrics on one manifold have the same geodesics. We show that then the geodesic flows of these metrics admit sufficiently many integrals (of a special form) in involution and vice versa. The quantum version of this result is also true. As a corollary we have that the topology of the manifold (with two different metrics with the same geodesics) must be sufficiently simple. Also we have that the non-proportionality of the metrics at a point induces the non-proportionality of the metrics at almost all points.

math.DG↗