  The paper is devoted to a local heat kernel, which is a special part of the
standard heat kernel. Locality means that all considerations are produced in an
open convex set of a smooth Riemannian manifold. We study such properties and
concepts as uniqueness, a symmetry of the Seeley-DeWitt coefficients, extension
on the whole manifold, a family of special functions, and the late-time
asymptotics with the use of the path integral formulation.
