Riemannian metric and riemannian manifold

Depends on

• A smooth tensor field
• Smooth bundle metrics

Used by

• A Morse gradient zero contributes (-1)^λ to the index Corollary
• Polar form of the metric in normal coordinates Corollary
• Polar integration may discard the cut locus Corollary
• A conjugate point at which there are many geodesics Counterexample
• Positive sectional curvature with no fixed lower bound on a noncompact manifold Counterexample
• Ricci lower bound does not control every sectional curvature in dimension at least three Counterexample
• Compact Kähler and rational Hodge polarization conventions Definition
• Complex manifolds, holomorphic vector bundles and Hermitian metrics Definition
• Complex Stiefel and Grassmann optimization conventions Definition
• Conformal equivalence of riemannian metrics Definition
• Conjugate points along a geodesic and their multiplicity Definition
• Geodesic triangulation with prescribed boundary arcs Definition
• H-one Riemannian curves and their half-energy Definition
• Hermitian metric and L² pairing on a compact Riemann surface Definition
• Injective geodesic ray transform on a transversal manifold Definition
• Musical isomorphisms Definition
• Oriented Riemannian surface and positive quarter-turn Definition
• Pointwise norm and angle from a riemannian metric Definition
• Pullback of a riemannian metric as a tensor Definition
• Real Stiefel and Grassmann optimization conventions Definition
• Riemann curvature four-tensor Definition
• Scalar conductivity and Schrödinger Dirichlet-to-Neumann maps Definition
• Simple Riemannian background Definition
• Simple surface and geodesic X-ray transform Definition
• Sobolev spaces on a compact Riemannian domain with boundary Definition
• Transversally anisotropic product domain Definition
• A conjugate antipode breaks global Bolker injectivity Example
• Cartan hadamard for hyperbolic space Example
• Conjugate antipodes on the round sphere Example
• Cut locus of a point on a flat circle Example
• Cut locus of a point on a round sphere Example
• Cut locus on a flat rectangular torus from the Dirichlet cell Example
• Distance hessian in euclidean space Example
• Dolbeault cohomology is independent of hermitian metric Example
• Exact-solution correction errors vanish in the Gaussian-beam ray limit Example
• Flat torus and zero Euler characteristic Example
• Flat torus dolbeault harmonic representatives Example
• Great circles as round-sphere geodesics Example
• Index form in constant curvature Example
• Killing jacobi fields from rotations Example

…and 63 more results.

Dependency tree · two levels

6 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

• John M. Lee, Introduction to Smooth Manifolds, second edition (standard reference, not scraped)
• Ved Datar, Lectures on Riemannian Geometry, September 2025 (standard reference, not scraped)