1306.6817 (Andrea Santi)

On Immersible G-Structures    [PDF]

Andrea Santi

The paper revisits and extends the theory of induced G-structures introduced by A. A. Rosly and A. S. Schwarz. Let N be a n-dimensional smooth manifold endowed with a H-structure, i.e. a reduction p:Q-->N of the principal bundle F_N of all linear frames on N to a Lie subgroup H of GL_n(R). Any m-dimensional submanifold M of N, satisfying fairly general regularity conditions, inherits a reduction \pi:P-->M of F_M to a Lie subgroup G of GL_m(R), called the G-structure induced by the ambient geometry (N,Q). We estabilish necessary and sufficient conditions for a G-structure on a manifold M to be locally equivalent to the G-structure induced by an homogeneous ambient geometry (N,Q)=(\tilde{H}/\tilde{K},\tilde{H}/K), where K denotes the kernel of the isotropy representation i:\tilde{K}-->GL_n(R). In the special case of integrable ambient geometry (R^n;R^nxH), the obstructions to constructing local equivalences are shown to be functions with values in the cohomology groups H^{p,2}_{R}(h) of a "restricted" Spencer cochain complex. Several examples are described in detail.

View original: http://arxiv.org/abs/1306.6817