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This is a glossary of terms specific to differential geometry and differential topology. The following three glossaries are closely related:

- Glossary of general topology
- Glossary of algebraic topology
- Glossary of Riemannian and metric geometry.

See also:

Words in *italics* denote a self-reference to this glossary.

**Bundle**, see *fiber bundle*.

A **basic element** *x* with respect to an element *y* is an element of a cochain complex (e.g., complex of differential forms on a manifold) that is closed: and the contraction of *x* by *y* is zero.

**Codimension**. The codimension of a submanifold is the dimension of the ambient space minus the dimension of the submanifold.

**Cotangent bundle**, the vector bundle of cotangent spaces on a manifold.

**Diffeomorphism.** Given two differentiable manifolds
*M* and *N*, a bijective map from *M* to *N* is called a **diffeomorphism** if both and its inverse 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.

**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}.

**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.

Hypersurface. A hypersurface is a submanifold of *codimension* one.

**Lens space**. A lens space is a quotient of the 3-sphere (or (2*n* + 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

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

**Parallelizable**. A smooth manifold is parallelizable if it admits a smooth global frame. This is equivalent to the tangent bundle being trivial.

**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.

**Submanifold**, the image of a smooth embedding of a manifold.

**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*.

**Transversality**. Two submanifolds *M* and *N* intersect transversally if at each point of intersection *p* their tangent spaces and 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 induces a vector bundle over *B* called the Whitney sum of these vector bundles and denoted by α⊕β.