Levi civita connection

Depends on

• Affine connection on a smooth manifold
• Metric compatible connection on a riemannian vector bundle
• Torsion tensor of an affine connection

Used by

• Maximal compact subgroups exist and are conjugate in a connected finite center semisimple Lie group Corollary
• Squared distance is strictly convex along geodesics in a hadamard manifold Corollary
• A conjugate point at which there are many geodesics Counterexample
• Conjugate points along a geodesic and their multiplicity Definition
• Connection one-form of an oriented orthonormal frame Definition
• Geodesic triangulation with prescribed boundary arcs Definition
• Induced connection and second fundamental form Definition
• Normal connection Definition
• Radial Jacobi tensor Definition
• Signed geodesic curvature Definition
• Simple Riemannian background Definition
• Simple surface and geodesic X-ray transform Definition
• Conjugate antipodes on the round sphere Example
• Jacobi fields in constant sectional curvature Example
• No conjugate points in nonpositive constant curvature Example
• Simple disk, trapped annulus, and conjugate-point cancellation Example
• Every affine connection is the levi civita connection of a riemannian metric False statement
• A complete no-return extension of a strictly convex manifold Lemma
• Global geodesic-polar coordinates from a boundary point on a simple manifold Lemma
• Integration by parts for the index form Lemma
• Koszul formula is necessary for a levi civita connection Lemma
• Local length comparison for a conjugate-free geodesic Lemma
• Surface geodesic flow, index form, and transport primitive Lemma
• Wronskian of two jacobi fields is constant Lemma
• Geodesic X-ray injectivity beyond the absence of conjugate points Open problem
• Cartan decomposition gives the invariant metric and curvature of G mod K Proposition
• Compact Lie groups admit bi-invariant metrics Proposition
• First variation of volume for a normal variation Proposition
• Hessian of distance in terms of radial jacobi fields Proposition
• Tangential jacobi fields are affine multiples of the velocity Proposition
• A geodesic does not minimize past its first conjugate point Theorem
• A Riemannian manifold is flat iff it is locally isometric to Euclidean space Theorem
• Algebraic symmetries of the Riemann tensor Theorem
• Contracted second Bianchi identity Theorem
• Differential of the exponential map in terms of Jacobi fields Theorem
• Differential second Bianchi identity Theorem
• Every Jacobi field is induced by a geodesic variation Theorem
• Existence and uniqueness of jacobi fields from initial data Theorem
• First Bianchi identity Theorem
• First variation formula for energy Theorem

…and 12 more results.

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Ved Datar, Lectures on Riemannian Geometry (standard reference, not scraped)