Smooth measures on manifolds with a Riemann- ian structure

The article is devoted to the current and actively developing direction of
development of the analysis of smooth measures on smooth infinite-dimensional manifolds. The

importance of this direction is dictated by the vast area of its applications, which include infinite-
dimensional analysis and a number of sections of mathematical physics of systems with an infinite

number of degrees of freedom.
In infinite-dimensional analysis, due to the absence of a standard measure such as Lebesgue
measure, the spaces of measures play the same role as the spaces of functions. Therefore, in
each of these spaces, a differential calculus must be constructed independently. Moreover, the
dual object to smooth measures is generalized functions, and to smooth functions - generalized
measures.

The differential calculus of measures in linear spaces was developed in the works of S.V.
Fomin, A.V. Averbukh, O.G. Smolyanov, as well as in the works of A.V. Skorokhod. But in all
these works, differentiation of measures in constant directions was considered. Therefore, it is
necessary to consider the concept of a derivative measure along a vector field, which makes sense
both in linear space and on a smooth manifold.

We establish the invariance of the derivative measure along the vector field with respect
to smooth transformations and the stability of the logarithmic derivative of the measure with
respect to smooth invertible mappings. Moreover, the derivative of the measure along the finite
set of vector fields turns out to be symmetric with respect to the vector fields. The corresponding
constructions lead to a generalization to non-Gaussian cases of integration formulas by parts,
which underlie the construction of extended stochastic integrals.

Covariant differential operations on manifolds with a Hilbert-Schmidt structure are
considered. The model of such a manifold is the Banach rigging of a real separable Hilbert space.
Partial integration formulas for such varieties are closely related to the differential-geometric
characteristics of the manifold; they contain the Ricci tensor of the manifold.