Glossary of differential geometry and topology
This is a glossary of terms specific to differential geometry and differential topology. The following two glossaies are closely related: See also: Words in italics denote a self-reference to this glossary.
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Bundle, see fiber bundle.
Codimension. The codimension of a submanifold is the dimension the ambient space minus the dimension of the submanifold.
Cotangent bundle, the vector bundle of cotangent spaces on a manifold.
Diffeomorphism. Given two differentiable manifolds
M and N, a bijective map from M to N is called a diffeomorphism if both and its inverse are smooth.
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.
Fiber. In a fiber bundle, π: E → B the preimage π−1(x) of a point x in the base B is called the fiber over x, often denoted Ex.
Frame
Frame bundle, the principal bundle of frames on a smooth manifold.
Flow
Hypersurface. A hypersurface is a submanifold of codimension one.
Lens space. A lens space is a quotient of the 3-sphere (or (2n+1)-sphere) by a free isometric action of Zk.
Manifold. A topological manifold is a locally Eulidean Hausdorff space. (In Wikipedia, a manifold need not be paracompact or second countable.) A Ck 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.
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 → B together with right action on P by a Lie group G that preverses the fibers of P and acts simply transitively on those fibers.
Submanifold. A submanifold is the image of a smooth embedding of a manifold.
Surface, a two-dimensional manifold or submanifold.
Tangent bundle, the vector bundle of tangent spaces on a differtiable 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 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 α and β over the same base B their cartesian product is a vector bundle over B ×B. The diagonal map induces a vector bundle over B called the Whitney sum of these vector bundles and denoted by α⊕β.
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