Sub-Riemannian Geometry

Inhaltsverzeichnis

• The tangent space in sub-Riemannian geometry.
• § 1. Sub-Riemannian manifolds.
• § 2. Accessibility.
• § 3. Two examples.
• § 4. Privileged coordinates.
• § 5. The tangent nilpotent Lie algebra and the algebraic structure of the tangent space.
• § 6. Gromov’s notion of tangent space.
• § 7. Distance estimates and the metric tangent space.
• § 8. Why is the tangent space a group?.
• References.
• Carnot-Carathéodory spaces seen from within.
• § 0. Basic definitions, examples and problems.
• § 1. Horizontal curves and small C-C balls.
• § 2. Hypersurfaces in C-C spaces.
• § 3. Carnot-Carathéodory geometry of contact manifolds.
• § 4. Pfaffian geometry in the internal light.
• § 5. Anisotropic connections.
• Survey of singular geodesics.
• § 1. Introduction.
• § 2. The example and its properties.
• § 3. Some open questions.
• § 4. Note in proof.
• A cornucopia of four-dimensional abnormal sub-Riemannian minimizers.
• § 2. Sub-Riemannian manifolds and abnormal extremals.
• § 3. Abnormal extremals in dimension 4.
• § 4. Optimality.
• § 5. An optimality lemma.
• § 6. End of the proof.
• § 7. Strict abnormality.
• § 8. Conclusion.
• Stabilization of controllable systems.
• § 0. Introduction.
• § 1. Local controllability.
• § 2. Sufficient conditions for local stabilizability of locally controllable systems by means of stationary feedback laws.
• § 3. Necessary conditions for local stabilizability by means of stationary feedback laws.
• § 4. Stabilization by means of time-varying feedback laws.
• § 5. Return method and controllability.