# 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 ${\displaystyle x}$  with respect to an element ${\displaystyle y}$  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 ${\displaystyle x}$  by ${\displaystyle y}$  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 ${\displaystyle M}$  and ${\displaystyle N}$ , a bijective map ${\displaystyle f}$  from ${\displaystyle M}$  to ${\displaystyle N}$  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 ${\displaystyle M}$  with boundary, doubling is taking two copies of ${\displaystyle M}$  and identifying their boundaries. As the result we get a manifold without boundary.

## F

• Fiber – In a fiber bundle, ${\displaystyle \pi :E\to B}$  the preimage ${\displaystyle \pi ^{-1}(x)}$  of a point ${\displaystyle x}$  in the base ${\displaystyle B}$  is called the fiber over ${\displaystyle x}$ , often denoted ${\displaystyle E_{x}}$ .
• 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 ${\displaystyle C^{k}}$  manifold is a differentiable manifold whose chart overlap functions are k times continuously differentiable. A ${\displaystyle C^{\infty }}$  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 ${\displaystyle P\to B}$  together with an action on ${\displaystyle P}$  by a Lie group ${\displaystyle G}$  that preserves the fibers of ${\displaystyle P}$  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 ${\displaystyle M}$  and ${\displaystyle N}$  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

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