Curvature: a variational approach

Abstract

The curvature discussed in this paper is a rather far going generalization of the Riemannian sectional curvature. We define it for a wide class of optimal control problems: a unified framework including geometric structures such as Riemannian, sub-Riemannian, Finsler and sub-Finsler structures; a special attention is paid to the sub-Riemannian (or Carnot-Caratheodory) metric spaces. Our construction of the curvature is direct and naive, and it is similar to the original approach of Riemann. Surprisingly, it works in a very general setting and, in particular, for all sub-Riemannian spaces.

120 pages, 12 figures, (v2) minor revision; (v3) new sections on Finsler manifolds, slow growth distributions, Heisenberg group; (v4) major revision, new extended section on 3D contact structures, constant curvature, improved results about existence of ample geodesics on SR structures, 2 new appendices, many minor revisions; (v5) 1 new appendix, minor revisions

References in corpus (4)

• Sub-Riemannian curvature in contact geometry
• On Jacobi fields and canonical connection in sub-Riemannian geometry
• Comparison theorems for conjugate points in sub-Riemannian geometry
• On conjugate times of LQ optimal control problems

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• On conjugate times of LQ optimal control problems
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• Failure of curvature-dimension conditions on sub-Riemannian manifolds via tangent isometries
• Spectral multipliers and wave equation for sub-Laplacians: lower regularity bounds of Euclidean type
• How many geodesics join two points on a contact sub-Riemannian manifold?
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• Small time asymptotic on the diagonal for Hörmander's type hypoelliptic operators
• Curvature exponent and geodesic dimension on Sard-regular Carnot groups
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• Kolmogorov-Fokker-Planck operators in dimension two: heat kernel and curvature
• Unified synthetic Ricci curvature lower bounds for Riemannian and sub-Riemannian structures
• On sub-Riemannian geodesic curvature in dimension three
• The projective symplectic geometry of higher order variational problems: minimality conditions
• Volume of small balls and sub-Riemannian curvature in 3D contact manifolds
• On the lower bound of the curvature exponent on step-two Carnot groups