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:
 * Glossary of general topology
 * Glossary of algebraic topology
 * Glossary of Riemannian and metric geometry.

See also:
 * List of differential geometry topics

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

A

 * Atlas

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

 * Chart


 * Cobordism


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


 * Connected sum


 * Connection


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


 * Cotangent space

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.

E

 * Embedding

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 $$E_x$$.


 * Fiber bundle


 * 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.


 * Flow

G

 * Genus

H

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

I

 * Immersion


 * Integration along fibers

L

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

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.

O

 * Orientation of a vector bundle

P

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


 * Poincaré lemma


 * 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.


 * Pullback

S

 * Section


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


 * Submersion


 * 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.


 * Tangent space


 * Thom space


 * Torus


 * 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 $$\alpha \oplus \beta$$.