On the total absolute curvature of manifolds immersed in Riemannian manifolds. III (original) (raw)
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American Journal of Mathematics, 1972
an (n -fp) -dimensional Eiemannian manifold Nn+p. Let h be the second fundamental form of this immersion; it is a certain symmetric bilinear mapping T^XTx-* Tw-L for x ? M, where T^ is the tangent space of M at x and IVthe normal space to M at x. We let 8 denote the length of h, H the mean curvature vector of this immersion, and a the length of H. If a = 0 identically on M, then M is called a minimal submanifold of Nn+p. In the first paper of this series [1], the author proved that the integral of an satisfies
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The curvature discussed in this paper is a far reaching generalisation of the Riemannian sectional curvature. We give a unified definition of curvature which applies to a wide class of geometric structures whose geodesics arise from optimal control problems, including Riemannian, sub-Riemannian, Finsler and sub-Finsler spaces. Special attention is paid to the sub-Riemannian (or Carnot-Carathéodory) metric spaces. Our construction of curvature is direct and naive, and similar to the original approach of Riemann. In particular, we extract geometric invariants from the asymptotics of the cost of optimal control problems. Surprisingly, it works in a very general setting and, in particular, for all sub-Riemannian spaces. Contents Chapter 1. Introduction v 1.1. Structure of the paper viii 1.2. Statements of the main theorems viii 1.3. The Heisenberg group x Part 1. Statements of the results Chapter 2. General setting 2.1. Affine control systems 2.2. End-point map 2.3. Lagrange multipliers rule 2.4. Pontryagin Maximum Principle 2.5. Regularity of the value function Chapter 3. Flag and growth vector of an admissible curve 3.1. Growth vector of an admissible curve 3.2. Linearised control system and growth vector 3.3. State-feedback invariance of the flag of an admissible curve 3.4. An alternative definition Chapter 4. Geodesic cost and its asymptotics 4.1. Motivation: a Riemannian interlude 4.2. Geodesic cost 4.3. Hamiltonian inner product 4.4. Asymptotics of the geodesic cost function and curvature 4.5. Examples Chapter 5. Sub-Riemannian geometry 5.1. Basic definitions 5.2. Existence of ample geodesics 5.3. Reparametrization and homogeneity of the curvature operator 5.4. Asymptotics of the sub-Laplacian of the geodesic cost 5.5. Equiregular distributions 5.6. Geodesic dimension and sub-Riemannian homotheties 5.7. Heisenberg group 5.8. On the "meaning" of constant curvature Part 2. Technical tools and proofs Chapter 6. Jacobi curves 6.1. Curves in the Lagrange Grassmannian 6.2. The Jacobi curve and the second differential of the geodesic cost 6.3. The Jacobi curve and the Hamiltonian inner product 6.4. Proof of Theorem A 6.5. Proof of Theorem D Chapter 7. Asymptotics of the Jacobi curve: equiregular case 7.1. The canonical frame 7.2. Main result iii 7.3. Proof of Theorem 7.4 7.4. Proof of Theorem B 7.5. A worked out example: 3D contact sub-Riemannian structures Chapter 8. Sub-Laplacian and Jacobi curves 8.1. Coordinate lift of a local frame 8.2. Sub-Laplacian of the geodesic cost 8.3. Proof of Theorem C
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