  This is the first in a series on papers developing an arithmetic PDE analogue
of Riemannian geometry. The role of partial derivatives is played by Fermat
quotient operations with respect to several Frobenius elements in the absolute
Galois group of a $p$-adic field. Existence and uniqueness of geodesics and of
Levi-Civita and Chern connections are proved in this context. In a sequel to
this paper a theory of arithmetic Riemannian curvature and characteristic
classes will be developed.
