# 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:

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

## B

• Bundle – see fiber bundle.
• basic element – 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.

## 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, $\pi :E\to B$  the preimage $\pi ^{-1}(x)$  of a point $x$  in the base $B$  is called the fiber over $x$ , often denoted $E_{x}$ .
• Frame bundle – the principal bundle of frames on a smooth manifold.

## H

• Hypersurface – A hypersurface is a submanifold of codimension one.

## M

• 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 $C^{\infty }$  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 $P\to B$  together with an action on $P$  by a Lie group $G$  that preserves the fibers of $P$  and acts simply transitively on those fibers.

## 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 $\alpha$  and $\beta$  over the same base $B$  their cartesian product is a vector bundle over $B\times B$ . 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 $\alpha \oplus \beta$ .