Differential Geometry is the study of geometric structures on manifolds. Manifolds are spaces which locally look like Euclidean space and therefore, one can do calculus on manifolds by means of coordinate charts. Examples of manifolds include surfaces in 3-space, complex projective space, and matrix (Lie) groups (e.g. O(n), SU(n), SL(n), etc.). The way in which coordinate charts are pieced together give the manifold a differentiable structure or a complex structure.
Riemannian structures in which an inner product is specified on the tangent space at each point of the manifold. This allows us to measure length of curves, angles and most importantly, to define curvature.
Symplectic structures in which a closed skew-symmetric bilinear form is specified on the tangent space at each point of the manifold. This structure was inspired by considerations from classical mechanics (Poisson bracket, etc. ).
Kähler structures in which a Riemannian structure, symplectic structure and complex structure interact in a harmonious way. Algebraic manifolds admit a Kähler structure. Kähler manifolds are of tremendous importance in modern physics, particularly string theories.
Symmetric structures, including hyperbolic structures.
PDEs and Analysis on Infinite Dimensional Spaces in Differential Geometry
The space of maps between manifolds is studied by pulling each map 'tight' and then studying the space of 'tight' maps. This involves studying a nonlinear version of the heat equation. Development of singularities is a major area of research. This area also has significant interaction with stochastic analysis.
Special submanifolds (e.g. ones which minimize volume) satisfy the Euler-Lagrange equations of some functional.
The Atiyah-Singer Index theorem provides a deep link between invariants associated to a linear differential operator on a manifold and algebraic topological invariants. This has led to a general study of index theories, which includes the development of new cohomology theories and analysis on path and loop spaces.