Manifold – A topological manifold is a locally Euclidean Hausdorff space. (In Wikipedia, a manifold need not be paracompact or second-countable.) A manifold is a differentiable manifold whose chart overlap functions are k times continuously differentiable. A or smooth manifold is a differentiable manifold whose chart overlap functions are infinitely continuously differentiable.
Neat submanifold – A submanifold whose boundary equals its intersection with the boundary of the manifold into which it is embedded.
Transversality – Two submanifolds and intersect transversally if at each point of intersection p their tangent spaces and generate the whole tangent space at p of the total manifold.
Vector bundle – a fiber bundle whose fibers are vector spaces and whose transition functions are linear maps.
Vector field – a section of a vector bundle. More specifically, a vector field can mean a section of the tangent bundle.
Whitney sum – A Whitney sum is an analog of the direct product for vector bundles. Given two vector bundles and over the same base their cartesian product is a vector bundle over . The diagonal map induces a vector bundle over called the Whitney sum of these vector bundles and denoted by .