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...

mathematics, a double vector bundle is the combination of two compatible vector bundle structures, which contains in particular the tangent T E {\displaystyle...

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...

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...

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...

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...

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...

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...

(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...

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...

mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way...

inverse Jacobian. A tensor bundle is a fiber bundle where the fiber is a tensor product of any number of copies of the tangent space and/or cotangent space...

classical gauge theory where gauge fields are affine connections on the tangent bundle over a smooth manifold X {\displaystyle X} . For instance, these are...

large transmission line project may have several types of towers, with "tangent" ("suspension" or "line" towers, UK) towers intended for most positions...

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...

noncontractible loop. Tangent bundle – the vector bundle of tangent spaces on a differentiable manifold. Tangent field – a section of the tangent bundle. Also called...

differential geometry. A smooth manifold always carries a natural vector bundle, the tangent bundle. Loosely speaking, this structure by itself is sufficient only...

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:...

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...

{\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)}...

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...

vector bundles provide information about the underlying topological space. For example, the tangent bundle consists of the collection of tangent spaces...

well-defined at every point of the tangent bundle. Intuitively speaking, the exponential map takes a given tangent vector to the manifold, runs along...

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...

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...

2-sphere, the 3-sphere admits nonvanishing vector fields (sections of its tangent bundle). One can even find three linearly independent and nonvanishing vector...

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...

γ ′ ( s ) , {\displaystyle \mathbf {T} (s)=\gamma '(s),}    (the unit tangent) u ( s ) = u ( γ ( s ) ) , {\displaystyle \mathbf {u} (s)=\mathbf {u} (\gamma...

covariant indices, because it has parts that live in the tangent bundle as well as the cotangent bundle. A contravariant vector is one which transforms like...

setting, ( v , q ) {\displaystyle (v,q)} are local coordinates on the tangent bundle T M {\displaystyle T{\mathcal {M}}} of a manifold M {\displaystyle {\mathcal...