  The aim of the present paper is to lay the foundation for a theory of
Ehresmann structures in positive characteristic, generalizing the
Frobenius-projective and Frobenius-affine structures defined in the previous
work. This theory deals with atlases of \'{e}tale coordinate charts on
varieties modeled on homogenous spaces of algebraic groups, which we call
Frobenius-Ehresmann structures. These structures are compared with Cartan
geometries in positive characteristic, as well as with higher-dimensional
generalizations of dormant indigenous bundles. In particular, we investigate
the conditions under which these geometric structures are equivalent to each
other. Also, we consider the classification problem of Frobenius-Ehresmann
structures on algebraic curves. The latter half of the present paper discusses
the deformation theory of indigenous bundles in the algebraic setting. The
tangent and obstruction spaces of various deformation functors are computed in
terms of the hypercohomology groups of certain complexes. As a consequence, we
formulate and prove the Ehresmann-Weil-Thurston principle for
Frobenius-Ehresmann structures. This fact asserts that deformations of a
variety equipped with a Frobenius-Ehresmann structure are completely determined
by their monodromy crystals.
