arXiv subjects

Dimitri Alekseevsky

Publications and source records attributed to Dimitri Alekseevsky.

2 recordsLinked to original sources

arXiv1993

Differential geometry of $\frak g$-manifolds

An action of a Lie algebra $\frak g$ on a manifold $M$ is just a Lie algebra homomorphism $ζ:\frak g\to \frak X(M)$. We define orbits for such an action. In general the space of orbits $M/\frak g$ is not a manifold and even has a bad topology. Nevertheless for a $\frak g$-manifold with equidimensional orbits we treat such notions as connection, curvature, covariant differentiation, Bianchi identity, parallel transport, basic differential forms, basic cohomology, and characteristic classes, which generalize the corresponding notions for principal $G$-bundles. As one of the applications, we derive a sufficient condition for the projection $M\to M/\frak g$ to be a bundle associated to a principal bundle.

math.DG↗

arXiv1992

Characteristic classes for $G$-structures

Let $G\subset GL(V)$ be a linear Lie group with Lie algebra $\frak g$ and let $A(\frak g)^G$ be the subalgebra of $G$-invariant elements of the associative supercommutative algebra $A(\frak g)= S(\frak g^*)\otimes \La(V^*)$. To any $G$-structure $π:P\to M$ with a connection $ω$ we associate a homomorphism $μ_ω:A(\frak g)^G\to Ω(M)$. The differential forms $μ_ω(f)$ for $f\in A(\frak g)^G$ which are associated to the $G$-structure $π$ can be used to construct Lagrangians. If $ω$ has no torsion the differential forms $μ_ω(f)$ are closed and define characteristic classes of a $G$-structure. The induced homomorphism $μ'_ω:A(\g)^G\to H^*(M)$ does not depend on the choice of the torsionfree connection $ω$ and it is the natural generalization of the Chern Weil homomorphism.

math.DG↗