nLab isometry group

Context

Riemannian geometry

Riemannian geometry

Applications

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Definition

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 $\phi : X \to X$ that are isometries: which preserve the metric $g$ in that

$\phi^* g = g \,.$

Properties

Killing vectors

The Lie algebra of $Iso(X,g)$ is spanned by the Killing vectors of $(X,g)$.

Examples

Last revised on May 8, 2015 at 12:19:09. See the history of this page for a list of all contributions to it.