- Glossary of differential geometry and topology
This is a
glossaryof terms specific to differential geometryand differential topology. The following two glossaries are closely related:
Glossary of general topology
Glossary of Riemannian and metric geometry.
List of differential geometry topics
Words in "italics" denote a self-reference to this glossary.
Bundle, see "fiber bundle".
Cobordism Codimension. The codimension of a submanifold is the dimension of the ambient space minus the dimension of the submanifold. Connected sum Cotangent bundle, the vector bundle of cotangent spaces on a manifold. Cotangent space
Diffeomorphism. Given two differentiable manifolds "M" and "N", a bijective mapfrom "M" to "N" is called a diffeomorphism if both and its inverse 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.
Fiber. In a fiber bundle, π: "E" → "B" the
preimageπ−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 spaceat the point. 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 paracompactor 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 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.
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".
Tangent space Torus
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.
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 productis 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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