synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
A differential form $\omega \in \Omega_{dR}^p(G)$ on a Lie group $G$ is called left invariant if for every $g \in G$ it is invariant under the pullback of differential forms
along the left multiplication action
Analogously a form is right invariant if it is invariant under the pullback by right translations $R_g$.
More generally, given a differentiable (e.g. smooth) group action of $G$ on a differentiable (e.g. smooth) manifold $M$
then a differential form $\omega \in \Omega^p_{dR}(M)$ is called invariant if for all $g \in G$
This reduces to the left invariance (1) for $M = G$ and $\rho$ being the left multiplication action of $G$ on itself.
For a vector field $X$ one instead typically defines the invariance via the pushforward $(T L_g) X = (L_g)_* X$. Regarding that $L_g$ and $T_g$ are diffeomorphisms, both pullbacks and pushfowards (hence invariance as well) are defined for every tensor field; and the two requirements are equivalent.
See most textbooks on on Lie theory, e.g.:
Sigurdur Helgason, Differential geometry, Lie groups and symmetric spaces, Graduate Studies in Mathematics 34 (2001) [ams:gsm-34]
François Bruhat (notes by S. Ramanan): Lectures on Lie groups and representations of locally compact groups, Tata Institute Bombay (1958, 1968) [pdf, pdf]
See also:
page 89 (20 of 49) at MIT course on Lie groups (pdf 2)
MathOverflow: Construction of the Lie functor: left vs. right invariant vector fields on Lie groups and Lie groupoids [MO:q/178528]
Last revised on July 11, 2024 at 10:52:58. See the history of this page for a list of all contributions to it.