Riemannian volume density

Depends on

• Coordinate criterion for a riemannian metric
• Density bundle and smooth density fields

Used by

• Polar integration may discard the cut locus Corollary
• Complex manifolds, holomorphic vector bundles and Hermitian metrics Definition
• E-valued Dolbeault operators and their L2 structure Definition
• Hermitian metric and L² pairing on a compact Riemann surface Definition
• Scalar conductivity and Schrödinger Dirichlet-to-Neumann maps Definition
• A conformal metric on the plane Example
• Exact-solution correction errors vanish in the Gaussian-beam ray limit Example
• Projective-plane curvature via a hemisphere Example
• The hyperbolic upper half space metric Example
• Volume density in polar coordinates Example
• Bishop gromov volume ratio is nondecreasing under a ricci lower bound False statement
• The laplace beltrami definition licenses the use of all euclidean harmonic function theory on manifolds False statement
• Alessandrini identities for scalar conductivity and Schrödinger DN maps Lemma
• Alessandrini identity for the time-dependent wave Dirichlet-to-Neumann maps Lemma
• An isometry preserves the finite hyperbolic volume integral Lemma
• Cayley Charts on the Unitary Matrix Group Lemma
• CTA Gaussian-beam quasimodes and geodesic concentration Lemma
• Finite propagation for the wave equation on a Riemannian manifold Lemma
• Gaussian beams concentrated on an embedded null bicharacteristic Lemma
• Riemannian Green identity, boundary trace, and wave lifting Lemma
• Smooth boundary tests separate locally integrable traces Lemma
• Summing local Gauss-Bonnet over a supplied triangulation Lemma
• Surface geodesic flow, index form, and transport primitive Lemma
• The Gauss-Bonnet expression is independent of the metric Lemma
• The riemannian volume density is coordinate independent Lemma
• Third linearization identity for the cubic Dirichlet-to-Neumann map Lemma
• Smooth anisotropic Calderón uniqueness from one boundary patch Open problem
• First variation of volume for a normal variation Proposition
• Levi civita parallel transport preserves lengths angles and volume Proposition
• Rigidity in bishop gromov on an interval Proposition
• Mean curvature and minimal submanifolds Remark
• A boundary-fixing diffeomorphism preserves the anisotropic DN map Theorem
• Bishop gromov volume comparison Theorem
• Cheng maximal diameter rigidity Theorem
• Cut locus of a point has riemannian volume zero Theorem
• Energy well-posedness for the variable-metric wave equation Theorem
• Gauss-Bonnet for closed nonorientable surfaces Theorem
• Harmonic representatives for compact de Rham and Dolbeault complexes Theorem
• Unweighted scalar geodesic X-ray injectivity on simple surfaces Theorem

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)