Glossary of differential geometry and topology
This is a glossary of terms specific to differential geometry and differential topology. The following three glossaries are closely related: See also: Words in "italics" denote a self-reference to this glossary. A. Atlas B. Bundle, see "fiber bundle". A basic element "x" with respect to an element "y" is an element of a cochain complex formula_1 (e.g., complex of differential forms on a manifold) that is closed: formula_2 and the contraction of "x" by "y" is zero. C. Chart Cobordism Codimension. The codimension of a submanifold is the dimension of the ambient space  minus the dimension of the submanifold. Connected sum Connection Cotangent bundle, the vector bundle of cotangent spaces on a manifold. Cotangent space D. Diffeomorphism. Given two differentiable manifolds "M" and "N", a bijective map formula_3 from "M" to "N" is called a diffeomorphism if both formula_4 and its inverse formula_5 are smooth functions. Doubling, given a manifold "M" with boundary, doubling is taking two copies of "M" and identifying their boundaries. As the result we get a manifold without boundary. E. Embedding F. Fiber. In a fiber bundle, π: "E" → "B" the preimage π("x") of a point "x" in  the base "B" is called the fiber over "x", often denoted "E". Fiber bundle Frame. A frame at a point of a differentiable manifold "M" is a basis of the tangent space at the point. Frame bundle, the principal bundle of frames on a smooth manifold. Flow G. Genus H. Hypersurface. A hypersurface is a submanifold of "codimension" one. I. Immersion Integration along fibers L. Lens space. A lens space is a quotient of the 3-sphere (or (2"n" + 1)-sphere) by a free isometric action of Z. M. Manifold. A topological manifold is a locally Euclidean Hausdorff space. (In  Wikipedia, a manifold need not be paracompact or second-countable.) A "C" manifold is a differentiable manifold whose chart overlap functions are "k" times continuously differentiable. A "C" or smooth manifold is a differentiable manifold whose chart overlap functions are infinitely continuously differentiable. N. Neat submanifold. A submanifold whose boundary equals its intersection with the boundary of the manifold into which it is embedded. O. Orientation of a vector bundle P. Parallelizable. A smooth manifold is parallelizable if it admits a smooth global frame. This is equivalent to the tangent bundle being trivial. Poincaré lemma Principal bundle. A principal bundle  is a fiber bundle "P" → "B" together with an action on "P" by a Lie group "G" that preserves the fibers of "P" and acts simply transitively on those fibers. Pullback S. Section Submanifold, the image of a smooth embedding of a manifold. Submersion Surface, a two-dimensional manifold or submanifold. Systole, least length of a noncontractible loop. T. Tangent bundle, the vector bundle of tangent spaces on a differentiable manifold. Tangent field, a "section" of the tangent bundle. Also called a "vector field". Tangent space Thom space Torus Transversality. Two submanifolds "M" and "N" intersect transversally if at  each point of intersection "p" their tangent spaces formula_6 and formula_7 generate the whole tangent space at "p" of the total manifold. Trivialization V. 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. W. Whitney sum. A Whitney sum is an analog of the direct product for vector bundles. Given two vector bundles α and β over the same base "B" their cartesian product is a vector bundle over "B" ×"B . The diagonal map formula_8 induces a vector bundle over B" called the Whitney sum of these vector bundles and denoted by α⊕β.