In mathematics, particularly differential topology, 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,M) of a smooth manifold M
.[1] A note on notation: in this article, we denote projection maps by their domains, e.g., πTTM : TTM → TM. Some authors index these maps by their ranges instead, so for them, that map would be written πTM.
The second tangent bundle arises in the study of connections and second order ordinary differential equations, i.e., (semi)spray structures on smooth manifolds, and it is not to be confused with the second order jet bundle.
^J.M.Lee, Introduction to Smooth Manifolds, Springer-Verlag, 2003.
and 24 Related for: Double tangent bundle information
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 tangentbundle of a smooth vector bundle (E...
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 doubletangentbundle TTM into horizontal and vertical bundles: T T M = H ⊕ V . {\displaystyle TTM=H\oplus...
can also be expressed in terms of the tangentbundle. The tangentbundle is a vector bundle, so it is a fiber bundle with structure group GL(n, R). That...
O(n)} . The example also works for bundles other than the tangentbundle; 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...
classical gauge theory where gauge fields are affine connections on the tangentbundle over a smooth manifold X {\displaystyle X} . For instance, these are...
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...
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 tangentbundle to the cotangent bundle. This section has the same...
noncontractible loop. Tangentbundle – the vector bundle of tangent spaces on a differentiable manifold. Tangent field – a section of the tangentbundle. Also called...
differential geometry. A smooth manifold always carries a natural vector bundle, the tangentbundle. Loosely speaking, this structure by itself is sufficient only...
frame or circle bundles of M. The definitions of the tangentbundle, the unit tangentbundle and the (oriented orthonormal) frame bundle F can be extended...
its tangentbundle TM.) The bundle of spinors πS: S → M over M is then the complex vector bundle associated with the corresponding principal bundle πP:...
vector bundles provide information about the underlying topological space. For example, the tangentbundle consists of the collection of tangent spaces...
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...
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...
covariant indices, because it has parts that live in the tangentbundle as well as the cotangent bundle. A contravariant vector is one which transforms like...
γ ′ ( s ) , {\displaystyle \mathbf {T} (s)=\gamma '(s),} (the unit tangent) u ( s ) = u ( γ ( s ) ) , {\displaystyle \mathbf {u} (s)=\mathbf {u} (\gamma...
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...
surface with elliptical directrix, the right conoid, the helicoid, and the tangent developable of a smooth curve in space. A ruled surface can be described...