- Glossary of differential geometry and topology
This is a

glossary of terms specific todifferential geometry anddifferential topology . The following two glossaries are closely related:

*Glossary of general topology

*Glossary of Riemannian and metric geometry .See also:

*List of differential geometry topics Words in "italics" denote a self-reference to this glossary.

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**A****B****Bundle**, see "fiber bundle".**C****Chart**Cobordism . The codimension of a submanifold is the dimension of the ambient space minus the dimension of the submanifold.Codimension Connected sum , the vector bundle of cotangent spaces on a manifold.Cotangent bundle Cotangent space **D**Given two differentiable manifolds "M" and "N", aDiffeomorphism .bijective map $f$ from "M" to "N" is called a**diffeomorphism**if both $f:M\; o\; N$ and its inverse $f^\{-1\}:N\; o\; M$ aresmooth function s.**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, π: "E" → "B" thepreimage π^{−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 adifferentiable manifold "M" is a basis of thetangent space at the point., the principal bundle of frames on a smooth manifold.Frame bundle **Flow****G****H****Hypersurface**. A hypersurface is a submanifold of "codimension" one.**I****L**. A lens space is a quotient of theLens space 3-sphere (or (2"n" + 1)-sphere) by a free isometric action of**Z**_{k}.**M**. A topological manifold is a locally EuclideanManifold Hausdorff space . (In Wikipedia, a manifold need not beparacompact or second-countable.) A "C^{k}" 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.**P**. A smooth manifold is parallelizable if it admits a smooth global frame. This is equivalent to the tangent bundle being trivial.Parallelizable . A principal bundle is a fiber bundle "P" → "B" together with right action on "P" by aPrincipal bundle Lie group "G" that preserves the fibers of "P" and acts simply transitively on those fibers.Pullback **Submanifold**. A submanifold is the image of a smooth embedding of a manifold., a two-dimensional manifold or submanifold.Surface **Systole**, least length of a noncontractible loop.**T**, the vector bundle of tangent spaces on a differentiable manifold.Tangent bundle **Tangent field**, a "section" of the tangent bundle. Also called a "vector field".Tangent 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**, a fiber bundle whose fibers are vector spaces and whose transition functions are linear maps.Vector bundle , a section of a vector bundle. More specifically, a vector field can mean a section of the tangent bundle.Vector field **W**. 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" theirWhitney sum cartesian product is a vector bundle over "B" ×"B". The diagonal map $B\; o\; B\; imes\; B$ induces a vector bundle over "B" called the Whitney sum of these vector bundles and denoted by α⊕β.

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