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The aim of the present paper is to lay the foundation for a theory ofEhresmann structures in positive characteristic, generalizing theFrobenius-projective and Frobenius-affine structures defined in the previouswork. This theory deals with atlases of \'{e}tale coordinate charts onvarieties modeled on homogenous spaces of algebraic groups, which we callFrobenius-Ehresmann structures. These structures are compared with Cartangeometries in positive characteristic, as well as with higher-dimensionalgeneralizations of dormant indigenous bundles. In particular, we investigatethe conditions under which these geometric structures are equivalent to eachother. Also, we consider the classification problem of Frobenius-Ehresmannstructures on algebraic curves. The latter half of the present paper discussesthe deformation theory of indigenous bundles in the algebraic setting. Thetangent and obstruction spaces of various deformation functors are computed interms of the hypercohomology groups of certain complexes. As a consequence, weformulate and prove the Ehresmann-Weil-Thurston principle forFrobenius-Ehresmann structures. This fact asserts that deformations of avariety equipped with a Frobenius-Ehresmann structure are completely determinedby their monodromy crystals.

Frobenius-Ehresmann structures and Cartan geometries in positive characteristic