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Glossary Of Differential Geometry And TopologyThis is a glossary of terms specific to differential geometry and differential topology. The following two glossaries are closely related: See also: Words in italics denote a self-reference to this glossary. A Atlas B Bundle, see fiber bundle. 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 from 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. E Embedding F 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. Fiber bundle Frame 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 L Lens space. A lens space is a quotient of the 3-sphere (or (2n+1)-sphere) by a free isometric action of Zk. M 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. P 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 S Section Submanifold. A submanifold is the image of a smooth embedding of a manifold. Submersion Surface, a two-dimensional manifold or submanifold. 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 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. 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 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. Differential geometry
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