Arithmetic Levi-Civita Connection (after A. Buium)

The theory of arithmetic jet spaces has been developed by A. Buium using rings with a delta structure for a fixed prime p. In this arithmetic setting, the delta structure on a ring plays the analogous role of a derivative operator in geometry. One naturally associates a lift of Frobenius morphism to such a delta structure. As an example, the delta structure on the ring of integers is the Fermat quotient operator and the identity map is the associated lift of Frobenius mod p.

In this talk, we discuss a recent paper by A. Buium where he introduces the arithmetic analogue of the differential geometric concept of the Levi-Civita connection. This talk will be self-contained introducing all the preliminary concepts.