Abstract Differentiable manifolds and their geometry appear naturally in the study of diverse areas of mathematics, including Lie group theory, homogeneous spaces, probability theory, differential equations, the theory of functions of a single complex variable and of several complex variables, algebraic geometry, classical mechanics, relativity theory, and the theory of elementary particles of physics. Riemannian geometry is a special geometry associated with differentiable manifolds and has many applications to physics. It is also a natural meeting place for several branches of mathematics.