the double tangent bundle or the second tangent bundle refers to the tangent bundle (TTM,πTTM,TM) of the total space TM of the tangent bundle (TM,πTM...
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vector bundle structure refers to the natural vector bundle structure (TE, p∗, TM) on the total space TE of the tangent bundle of a smooth vector bundle (E...
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mathematics, a double vector bundle is the combination of two compatible vector bundle structures, which contains in particular the tangent T E {\displaystyle...
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to the manifold at that point. Tangent bundles are not, in general, trivial bundles. For example, the tangent bundle of the sphere is non-trivial by...
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a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at...
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connection gives rise to a splitting of the double tangent bundle TTM into horizontal and vertical bundles: T T M = H ⊕ V . {\displaystyle TTM=H\oplus...
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O(n)} . The example also works for bundles other than the tangent bundle; if E {\displaystyle E} is any vector bundle of rank k {\displaystyle k} over M...
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Orientability (redirect from Orientable double cover)
can also be expressed in terms of the tangent bundle. The tangent bundle is a vector bundle, so it is a fiber bundle with structure group GL(n, R). That...
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manifold and (TM,πTM,M) its tangent bundle. Then a vector field H on TM (that is, a section of the double tangent bundle TTM) is a semi-spray on M, if...
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(in terms of the tangent bundle, not stable normal bundle) by Whitney. For example, the Möbius strip has non-trivial tangent bundle, so it cannot immerse...
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Tensor field (section Twisting by a line bundle)
(V^{*})^{\otimes q})} where V = T M {\displaystyle V=TM} to be the tangent bundle of M {\displaystyle M} (whose sections are called vector fields or contra...
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mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way...
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noncontractible loop. Tangent bundle – the vector bundle of tangent spaces on a differentiable manifold. Tangent field – a section of the tangent bundle. Also called...
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Metric tensor (section Metric as a section of a bundle)
Sg defines a section of the bundle Hom(TM, T*M) of vector bundle isomorphisms of the tangent bundle to the cotangent bundle. This section has the same...
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Differential geometry (section Bundles and connections)
differential geometry. A smooth manifold always carries a natural vector bundle, the tangent bundle. Loosely speaking, this structure by itself is sufficient only...
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Vector space (section Vector bundles)
vector bundles provide information about the underlying topological space. For example, the tangent bundle consists of the collection of tangent spaces...
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{\displaystyle n} -form that vanishes nowhere. The structure group of the tangent bundle of M {\displaystyle M} can be reduced from U ( n ) {\displaystyle U(n)}...
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Overhead power line (redirect from Bundle conductor)
large transmission line project may have several types of towers, with "tangent" ("suspension" or "line" towers, UK) towers intended for most positions...
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its tangent bundle TM.) The bundle of spinors πS: S → M over M is then the complex vector bundle associated with the corresponding principal bundle πP:...
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Affine gauge theory (section Affine tangent bundle)
classical gauge theory where gauge fields are affine connections on the tangent bundle over a smooth manifold X {\displaystyle X} . For instance, these are...
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well-defined at every point of the tangent bundle. Intuitively speaking, the exponential map takes a given tangent vector to the manifold, runs along...
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γ ′ ( s ) , {\displaystyle \mathbf {T} (s)=\gamma '(s),} (the unit tangent) u ( s ) = u ( γ ( s ) ) , {\displaystyle \mathbf {u} (s)=\mathbf {u} (\gamma...
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general Riemannian and pseudo-Riemannian manifolds, one has a tangent bundle, a cotangent bundle and a metric that ties the two together. There are several...
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frame or circle bundles of M. The definitions of the tangent bundle, the unit tangent bundle and the (oriented orthonormal) frame bundle F can be extended...
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frame bundle so that its tangent vectors lie in a special subspace of codimension one in the three-dimensional tangent space of the frame bundle. The projection...
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homomorphism. Let E be a vector bundle on X of rank r and q: P(E ⊕ 1) → X the projective bundle (here 1 means the trivial line bundle). As usual, we identity...
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value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of...
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p orbitals overlap, called a pi bond. The natural projection on the tangent bundle on a manifold. The unary operation of projection in relational algebra...
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normal curve. 2. Orthogonal to the tangent space, such as a line orthogonal to the tangent space or the normal bundle. 3. A normal intersection is an intersection...
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2-sphere, the 3-sphere admits nonvanishing vector fields (sections of its tangent bundle). One can even find three linearly independent and nonvanishing vector...
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