Kemotsu Space Forms

In the early 19th century we find diverse steps towards a generalization of geometric language to higher dimensions. But they were still of a tentative and often merely metaphorical character. The analytical description of dynamical systems in classical mechanics was a field in which, from hindsight, one would expect a drive towards and a growing awareness of the usefulness of higher dimensional geometrical language. Manifolds, the higher-dimensional analogues of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory. Differential manifoldis the abstract generalization of smooth curves and surfaces in Euclidean space. Such a manifold has a topology and a certain dimension n,and locally it is homeomorphic with a piece of n-dimensional Euclidean space, such that these pieces are differentiably glued together.