Flat bundle

In mathematics , flat bundles occur in differential geometry and mathematical physics , among others .

definition

A shallow bundle is a principal bundle that has a shallow context . ${\displaystyle P\rightarrow M}$

A connection is called flat if its curvature disappears, i.e. if ${\displaystyle \omega }$ ${\displaystyle \Omega }$

${\displaystyle \Omega :=d\omega +{\tfrac {1}{2}}[\omega \wedge \omega ]=0.}$

Geometric interpretation

According to Ambrose-Singer's theorem, curvature measures infinitesimal holonomy . For a principal bundle with a shallow context, the holonomy must be infinitesimally (but not necessarily globally) trivial; H. homotopic ways have the same holonomy. In particular, holonomy induces a well-defined representation of the fundamental group of the base in the structural group of the principal bundle. ${\displaystyle \pi _{1}M\rightarrow G}$

Holonomy representation

Flat G-bundles over a connected manifold are in bijection with representations ${\displaystyle M}$

${\displaystyle \rho :\pi _{1}M\rightarrow G}$.

The flat bundle associated with a representation is obtained - with the aid of the action of on the universal overlay - as ${\displaystyle \pi _{1}M}$ ${\displaystyle {\widetilde {M}}}$

${\displaystyle E_{\rho }:={\widetilde {M}}\times G/\sim }$

with the equivalence relation for . ${\displaystyle (\gamma x,g)\sim (x,\rho (\gamma )g)}$ ${\displaystyle \gamma \in \pi _{1}M,x\in {\widetilde {M}},g\in G}$

Sections in clearly correspond to the -equivariant images , the corresponding section is for (any) to be projected . ${\displaystyle E_{\rho }}$ ${\displaystyle \rho }$ ${\displaystyle f\colon {\widetilde {M}}\to G}$ ${\displaystyle f}$ ${\displaystyle s(x)=\left[{\widetilde {x}},f({\tilde {x}})\right]}$ ${\displaystyle x\in M}$ ${\displaystyle {\tilde {x}}\in {\widetilde {M}}}$

literature

• Morita, Shigeyuki: Geometry of characteristic classes. Translated from the 1999 Japanese original. Translations of Mathematical Monographs, 199. Iwanami Series in Modern Mathematics. American Mathematical Society, Providence, RI, 2001. ISBN 0-8218-2139-3
• Kamber, Franz W .; Tondeur, Philippe: Foliated bundles and characteristic classes. Lecture Notes in Mathematics, Vol. 493. Springer-Verlag, Berlin-New York, 1975.