Riemannian geometry
Riemannian manifold
moduli space of Riemannian metrics
pseudo-Riemannian manifold
geodesic
geodesic convexity
geodesic flow
Levi-Civita connection
Hodge inner product, Hodge star operator
gradient, gradient flow
gravity
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differential geometry
synthetic differential geometry
smooth topos
infinitesimal space
Kock-Lawvere axiom
integration axiom
microlinear space
synthetic differential supergeometry
. infinitesimal cohesion
de Rham space
formally smooth morphism, formally unramified morphism, formally étale morphism
jet bundle
Models for Smooth Infinitesimal Analysis
Fermat theory
smooth algebra (C ∞-ring)
smooth locus
smooth manifold, formal smooth manifold, derived smooth manifold
smooth space, diffeological space, Frölicher space
smooth natural numbers
Cahiers topos
smooth ∞-groupoid
synthetic differential ∞-groupoid
tangent bundle,
vector field, tangent Lie algebroid;
differentiation, chain rule
differential forms
differential equation, variational calculus
connection on a bundle, connection on an ∞-bundle
Hadamard lemma
Borel's theorem
Boman's theorem
Steenrod-Wockel approximation theorem
Poincare lemma
Stokes theorem
de Rham theorem
Chern-Weil theory
Lie theory, ∞-Lie theory
Chern-Weil theory, ∞-Chern-Weil theory
gauge theory
∞-Chern-Simons theory
Klein geometry, Klein 2-geometry, higher Klein geometry
Euclidean geometry, Cartan geometry, higher Cartan geometry
For (X,g) a Riemannian manifold or pseudo-Riemannian manifold its isometry group
Iso(X,g) \subset Diff(X)
is that subgroup of the group of all diffeomorphisms ϕ:X→X that are isometries: which preserve the metric g in that
\phi^* g = g \,.
The Lie algebra of Iso(X,g) is spanned by the Killing vectors of (X,g).