In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It arises naturally from the study of the theory of differential equations. Differential geometry is the study of geometry using calculus. These fields are adjacent, and have many applications in physics, notably in the theory of relativity. Together they make up the geometric theory of differentiable manifolds - which can also be studied directly from the point of view of dynamical systems.
Intrinsic versus extrinsic
Initially and up to the middle of the nineteenth century, differential geometry was studied from the extrinsic point of view: curves, surfaces were considered as lying in a Euclidean space of higher dimension (for example a surface in an ambient space of three dimensions). The simplest results are those in the differential geometry of curves. Starting with the work of Riemann, the intrinsic point of view was developed, in which one cannot speak of moving 'outside' the geometric object because it is considered as given in a free-standing way.
The intrinsic point of view is more flexible, for example it is useful in relativity where space-time cannot naturally be taken as extrinsic. With the intrinsic point of view it is harder to define curvature and other structures such as connection, so there is a price to pay.
These two points of view can be reconciled, i.e. the extrinsic geometry can be considered as a structure additional to the intrinsic one (see the Nash embedding theorem).
The apparatus of differential geometry is that of calculus on manifolds: this includes the study of manifolds, tangent bundles, cotangent bundles, differential forms, exterior derivatives,integrals of p-forms over p-dimensional submanifolds and Stokes' theorem, wedge products, and Lie derivatives. These all relate to multivariate calculus; but for the geometric applications must be developed in a way that makes good sense without a preferred coordinate system. The distinctive concepts of differential geometry can be said to be those that embody the geometric nature of the second derivative: the many aspects of curvature.
A differential manifold is a topological space with a collection of homeomorphisms from open sets to the open unit ball in Rn such that the open sets cover the space, and if f, g are homeomorphisms then the function f-1 o g from an open subset of the open unit ball to the open unit ball is infinitely differentiable. We say a function from the manifold to R is infinitely differentiable if its composition with every homemorphism results in an infinitely differentiable function from the open unit ball to R.
At every point of the manifold, there is the tangent space at that point, which consists of every possible velocity (direction and magnitude) with which it is possible to travel away from this point. For an n-dimensional manifold, the tangent space at any point is an n-dimensional vector space, or in other words a copy of Rn. The tangent space has many definitions. One definition of the tangent space is as the dual space to the linear space of all functions which are zero at that point, divided by the space of functions which are zero and have a first derivative of zero at that point. Having a zero derivative can be defined by "composition by every differentiable function to the reals has a zero derivative", so it is defined just by differentiability.
A vector field is a function from a manifold to the disjoint union of its tangent spaces (this union is itself a manifold known as the tangent bundle), such that at each point, the value is an element of the tangent space at that point. Such a mapping is called a section of a bundle. A vector field is differentiable if for every differentiable function, applying the vector field to the function at each point yields a differentiable function. Vector fields can be thought of as time-independent differential equations. A differentiable function from the reals to the manifold is a curve on the manifold. This defines a function from the reals to the tangent spaces: the velocity of the curve at each point it passes through. A curve will be said to be a solution of the vector field if, at every point, the velocity of the curve is equal to the vector field at that point.
An alternating k-dimensional linear form is an element of the antisymmetric k'th tensor power of the dual V* of some vector space V. A differential k-form on a manifold is a choice, at each point of the manifold, of such an alternating k-form -- where V is the tangent space at that point. This will be called differentiable if whenever it operates on k differentiable vector fields, the result is a differentiable function from the manifold to the reals. A space form is a linear form with the dimensionality of the manifold.
Branches of differential geometry/topology
This is an analog of symplectic geometry which works for manifolds of odd dimension. Roughly, the contact structure on (2n+1)-dimensional manifold is a choice of a 1-form α such that does not vanish anywhere.
Finsler geometry has the Finsler manifold as the main object of study — this is a differential manifold with a Finsler metric, i.e. a Banach norm defined on each tangent space. A Finsler metric is much more general structure than a Riemannian metric.
Riemannian geometry has Riemannian manifolds as the main object of study — smooth manifolds with additional structure which makes them look infinitesimally like Euclidean space. These allow one to generalise the notion from Euclidean geometry and analysis such as gradient of a function, divergence, length of curves and so on; without assumptions that the space is globally so symmetric.
This is the study of symplectic manifolds. A symplectic manifold is a differentiable manifold equipped with a symplectic form (that is, a closed non-degenerate 2-form).
A Modern Course on Curves and Surface, Richard S Palais, 2003 (http://rsp.math.brandeis.edu/3D-XplorMath/Surface/a/bk/curves_surfaces_palais.pdf)
Richard Palais's 3DXM Surfaces Gallery (http://rsp.math.brandeis.edu/3D-XplorMath/Surface/gallery.html)