chapter  2
57 Pages

Analysis on Riemannian Manifolds

WithAleksei A. Dezin

The multidimensional analysis is an analysis on a Riemannian manifold, and a domain of the Euclidean space is one of the most important cases of the latter. It considers only the simplest elements of the Riemannian structure, not connected with curvature, or covariant differentiation, etc. At the same time, the term "analysis" is understood in a very wide sense since it includes the theory of differential equations and boundary value problems. However, the elements of the Riemannian structure being introduced prove to be sufficient for defining the classes of partial differential equations. It is natural to study these equations in the mentioned context. A special place will be occupied by the relationship between the formalism and the classical vector analysis and the standard equations of mathematical physics. The latter include the Laplace equation, the wave equation, the Cauchy-Riemann equations and their multidimensional generalizations, the Maxwell equations, and the Navier–Stokes equations.