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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*.

**basic element**– A**basic element***with respect to an element**is an element of a cochain complex (e.g., complex of differential forms on a manifold) that is closed: and the contraction of**by**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*and**, a bijective map from**to**is called a***diffeomorphism**– if both and its inverse are smooth functions.

**Doubling**– Given a manifold*with boundary, doubling is taking two copies of**and identifying their boundaries. As the result we get a manifold without boundary.*

**Fiber**– In a fiber bundle,*the preimage**of a point**in the base**is called the fiber over**, often denoted**.*

**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*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.

**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*together with an action on**by a Lie group**that preserves the fibers of**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*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.

**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*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**.*