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Associated bundle (topology) Hopf bundle (topology) I bundle (topology) Vector bundle (topology) Regroup S.C. Into Note (Vector Bundle) Principle bundle (topology) Frame bundle (topology) To move / sort Bundle (topology) Double tangent bundle. 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. Secondary vector bundle structure and canonical flip[edit] Since (TM,πTM,M) is a vector bundle in its own right, its tangent bundle has the secondary vector bundle structure (TTM,(πTM)*,TM), where (πTM)*:TTM→TM is the push-forward of the canonical projection πTM:TM→M.
In the following we denote and apply the associated coordinate system on TM. Then the fibre of the secondary vector bundle structure at X∈TxM takes the form The double tangent bundle is a double vector bundle. The canonical flip has the property that for any f: R2 → M, where α(s,t)= (t,s). Double vector bundle. From Wikipedia, the free encyclopedia In mathematics, a double vector bundle is the combination of two compatible vector bundle structures, which contains in particular the tangent of a vector bundle and the double tangent bundle Definition and first consequences[edit] A double vector bundle consists of , where the side bundles and are vector bundles over the base , is a vector bundle on both side bundles and ,the projection, the addition, the scalar multiplication and the zero map on E for both vector bundle structures are morphisms.
Double vector bundle morphism[edit] A double vector bundle morphism consists of maps and such that is a bundle morphism from to The 'flip of the double vector bundle is the double vector bundle Examples[edit] If is a vector bundle over a differentiable manifold then is a double vector bundle when considering its secondary vector bundle structure. is a differentiable manifold, then its double tangent bundle is a double vector bundle. References[edit] Secondary vector bundle structure. From Wikipedia, the free encyclopedia In the special case (E, p, M) = (TM, πTM, M), where TE = TTM is the double tangent bundle, the secondary vector bundle (TTM, (πTM)∗, TM) is isomorphic to the tangent bundle(TTM, πTTM, TM) of TM through the canonical flip. Construction of the secondary vector bundle structure[edit] Let (E, p, M) be a smooth vector bundle of rank N. Then the preimage (p∗)−1(X) ⊂ TE of any tangent vector X in TM in the push-forward p∗ : TE → TM of the canonical projection p : E → M is a smooth submanifold of dimension 2N, and it becomes a vector space with the push-forwards of the original addition and scalar multiplication as its vector space operations.
Proof[edit] Let (U, φ) be a local coordinate system on the base manifold M with φ(x) = (x1, ..., xn) and let be a coordinate system on adapted to it. So the fiber of the secondary vector bundle structure at X in TxM is of the form Now it turns out that and Linearity of connections on vector bundles[edit] See also[edit] P.Michor. Topology (math)