Riemann curvature four-tensor

Depends on

• Curvature is a type (1,3) tensor
• Riemannian metric and riemannian manifold
• Fundamental theorem of riemannian geometry

Used by

• Bonnet-Myers fundamental group is finite Corollary
• Diameter rigidity from toponogov under a sectional lower bound Corollary
• Upper sectional curvature bounds delay conjugate points Corollary
• Ricci lower bound does not control every sectional curvature in dimension at least three Counterexample
• Index form of a geodesic segment Definition
• Jacobi field Definition
• Radial Jacobi tensor Definition
• Ricci curvature Definition
• Sectional curvature Definition
• A deformed sphere has the same total curvature Example
• A flat torus showing simple connectedness is needed for global exp injectivity Example
• Bonnet-Myers for the round sphere Example
• Cartan hadamard for hyperbolic space Example
• Curvature of a Riemannian product Example
• Distance hessian and laplacian in space forms Example
• Gaussian curvature of a surface of revolution Example
• Hyperbolic space has negative constant sectional curvature Example
• The round two-sphere has zero untwisted Spin Dirac kernel Example
• Cartan hadamard says exp p is injective without simple connectedness False statement
• Positive ricci curvature without a uniform lower bound implies compactness False statement
• Ricci curvature and scalar curvature determine the full Riemann tensor in every dimension False statement
• Lichnerowicz formula for a geometric Dirac operator Lemma
• Spinor and geometric Clifford bundles and connections are well defined Lemma
• Surface geodesic flow, index form, and transport primitive Lemma
• Wronskian of two jacobi fields is constant Lemma
• Cartan decomposition gives the invariant metric and curvature of G mod K Proposition
• Flat torus model geometry Proposition
• Hessian of distance in terms of radial jacobi fields Proposition
• Jacobi fields are the null solutions of the index form with fixed endpoints Proposition
• Jacobi fields are the null solutions of the index form with fixed endpoints Proposition
• Rigidity in bishop gromov on an interval Proposition
• Rigidity in rauch comparison Proposition
• Upper half-space model geometry Proposition
• A Riemannian manifold is flat iff it is locally isometric to Euclidean space Theorem
• Algebraic symmetries of the Riemann tensor Theorem
• Bonnet conjugate radius theorem Theorem
• Bonnet myers Theorem
• Differential second Bianchi identity Theorem
• Existence and uniqueness of jacobi fields from initial data Theorem
• Gauss equation for a Riemannian submanifold Theorem

…and 7 more results.

Dependency tree · two levels

9 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)
• John M. Lee, Riemannian Manifolds: An Introduction to Curvature (standard reference, not scraped)