Differential geometry, convex bodies, integral geometry

Abstract

In differential geometry the concepts and methods of analysis, particularly of differential calculus and the theory of differential equations, are applied to the study of geometric figures. The underlying geometrical spaces or manifolds must, as in analytical geometry, be referred to coordinates. Other geometrical figures are embedded in these spaces, for example, general curves or curved surfaces, which are characterized by sufficiently differentiate equations or functions. To understand the more advanced parts of differential geometry, one must be fully conversant with the tensor calculus; furthermore, a knowledge of topology and some other branches of mathematics is essential.