A vector field is a function from a manifold to the disjoint union of its tangent spaces, such that at each point, the value is a member of the tangent space at that point. 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.
Subareas include Symplectic topology, the study of manifolds with non-degenerate bi-linear forms.