The fundamental constituents of geometry such as curves and surfaces in three dimensional space, lead us to the consideration of higher dimensional objects called manifolds. The manifolds which live in this higher dimensional world can be very complicated, and the subject of geometry expands wonderfully to accommodate them. Nevertheless, arguments and conclusions about these fantastic shapes retain the universal mathematical spirit of truth and clarity. Differential geometry begins by examining curves and surfaces, and the extend to which they are curved.
The precise mathematical definition of curvature can be made into a powerful toll for studying the geometrical structure of manifolds of higher dimensions.
Topology is concerned with the most basic underlying features of manifolds, when all geometrical concepts such as length and angle are ignored. Only the property of continuity is studied. For example, the different ways of making knots in a piece of string may be distinguished without reference to the length of the string or its diameter.
Manifolds are not simply a creation of mathematical imagination. They appear in practical problems as well, where they provide a meeting point for geometry, topology, analysis and various branches of applied mathematics and physics. Important examples of manifolds are Euclidean spaces, the sphere, the torus, projective spaces, Lie groups (spaces with additionally a group structure), and homogeneous spaces G/H (formal space of cosets). Difficult problems from geometry or differential equations can be handled by using techniques from Lie theory.