Canonical fibre bundles over orbit spaces, and intrinsic connections

Ü. G. Lumiste

Itogi Nauki i Tekhniki. Seriya Problemy Geometrii. Trudy Geometricheskogo Seminara, Trudy Geometricheskogo Seminara, Tome 4 (1973), pp. 285-307.

Résumé

A generalization of the well-known concept of canonical vector bundle over Grassmannian manifold is given. Let $\mathfrak G/\mathfrak H$ be a homogeneous space with a given point $X_0$ in it and let $G$ be a closed subgroup of its structural group $\mathfrak G$. Then the orbit $x_0=G_0\circ X_0$ is given in $\mathfrak G/\mathfrak H$. Each $g\in\mathfrak G$, acting in $\mathfrak G/\mathfrak H$, maps $x_0$ onto $x=g\circ x_0=(gGg^{-1})\circ(g\circ X_0)$, which is also an orbit. The set $M=\{g\circ x_0\mid g\in\mathfrak G\}$ is said to be a space of equivalent orbits with the representative $x_0$. It is shown that the set $E\subset\mathfrak G/\mathfrak H\times M$, defined by $E=\{(X,x)\mid X\in x\}$ has a natural structure of locally trivial analytical fibre bundle $p\colon E\to M$, where $p(X,x)=x$, with the homogeneous typical fibre $G/H$, where $H=\mathfrak H\cap G$. The bundle is called the canonical fibre bundle over $M$ for $\mathfrak G/\mathfrak H$. Its slight generalizations and a special case, called the incidence bundle, are given as well. If the homogeneous space $M$ is reductive, there is a one-one correspondence between the set of decompositions $\mathfrak G=G\oplus K$ with $[K,G]\subset K$ and the set of connections in $p\colon\overline E\to\overline M$. If there is only one such kind of decomposition, the correspondending connection is called intrinsic. An example is given: the intrinsic conformal connection in canonical fibre, bundle over space of hyperquadrics for projective space with its geometrical signification. In a slightlymore general case for $M$ the concept of induced connection in $p\colon E\to M$ and some of its examples with their geometrical significations are given.