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Glossary of differential geometry and topology ## Summary

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

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

## B

Bundle, see fiber bundle.

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

## C

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.

## D

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

## F

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

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.

## H

Hypersurface. A hypersurface is a submanifold of codimension one.

## L

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

## M

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

## P

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 PB together with an action on P by a Lie group G that preserves the fibers of P and acts simply transitively on those fibers.

## S

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

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.

Transversality. Two submanifolds M and N intersect transversally if at each point of intersection p their tangent spaces $T_{p}(M)$  and $T_{p}(N)$  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 $B\to B\times B$  induces a vector bundle over B called the Whitney sum of these vector bundles and denoted by α⊕β.