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:

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.


  • 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 bundle – the principal bundle of frames on a smooth manifold.



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




  • 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  .