Glossary of differential geometry and topology

Atlas

Bundle, see fiber bundle.

A basic element x with respect to an element y is an element of a cochain complex ${\displaystyle (C^{*},d)}$ (e.g., complex of differential forms on a manifold) that is closed: ${\displaystyle dx=0}$ and the contraction of x by y is zero.

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

Diffeomorphism. Given two differentiable manifolds M and N, a bijective map ${\displaystyle f}$ from M to N is called a diffeomorphism if both ${\displaystyle f:M\to N}$ and its inverse ${\displaystyle f^{-1}:N\to M}$ 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.

Embedding

Fiber. In a fiber bundle, π: E → B the preimage π⁻¹(x) of a point x in the base B is called the fiber over x, often denoted E_x.

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

Genus

Hypersurface. A hypersurface is a submanifold of codimension one.

Immersion

Integration along fibers

Lens space. A lens space is a quotient of the 3-sphere (or (2n + 1)-sphere) by a free isometric action of Z_k.

Manifold. A topological manifold is a locally Euclidean Hausdorff space. (In Wikipedia, a manifold need not be paracompact or second-countable.) A C^k 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.

Neat submanifold. A submanifold whose boundary equals its intersection with the boundary of the manifold into which it is embedded.

Orientation of a vector bundle

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

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.

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 ${\displaystyle T_{p}(M)}$ and ${\displaystyle T_{p}(N)}$ generate the whole tangent space at p of the total manifold.

Trivialization

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 α and β over the same base B their cartesian product is a vector bundle over B ×B. The diagonal map ${\displaystyle B\to B\times B}$ induces a vector bundle over B called the Whitney sum of these vector bundles and denoted by α⊕β.