Tangent bundle

The Semple bundle varieties were first introduced in [Semple (1)]. In [Collino (1)], [Colley-Kennedy (1),(2)] they are considered for arbitrary smooth surfaces. The construction of Fn(X) for an arbitrary smooth projective variety X is an obvious generalisation. For our purposes it appears to be slightly more practical to use the tangent bundles instead of the cotangent bundles in the construction. 

It is well known that the relative tangent bundle of a projectivized vector bundle E of rank r is... 

The tangent bundle T(M) associated with the configuration space M is a set of F-dimensional vector spaces TQ (fibres) with coordinates (velocities) Vj each fiber TQisa tangent space to the configuration space Mata point Q (42). Hence the tangent bundle T(M) is locally described by 2F coordinates (Q, V) = Q Vj. Since any transformation of coordinates Q in M induces a linear transformation of Vj in TQ (42), the solution to the equations of motion does not depend on the choice of the coordinate system in M. 

In other words, / is a sort of disembodied tangent vector which can be embedded in any scheme X so as to lie along any given tangent vector to X. The set of all morphisms from / to X is a sort of set-theoretic tangent bundle to X,... 

This is a system of linear equations for A. Assuming g ( )Af g ( ) is invertible, we may solve for A. Reinserting this expression for A into (4.8) then gives a system of ordinary differential equations whose solutions, starting from a point on T M, will remain on T M for all time. The co-tangent bundle becomes an invariant manifold of this ODE system. A is defined at any time by the physical variables q andp. 

Describing the Hamiltonian structure for a constrained system is a little complicated to do formally. The simplifying concept that we exploit is that the symplectic 2-form in the ambient space can be projected to the co-tangent bundle to define an associated symplectic form on the manifold. 

Thus we see that the projected version of the symplectic form in the ambient space is conserved by the differential equation system (4.7)-(4.9). This is what is meant by saying that the constrained system is analogous to a Hamiltonian system. We may also think of its flow map as being a symplectic map of the co-tangent bundle. 

It is difficult to make sense of this question, since (4.14)-(4.16) does not even constitute a map of the co-tangent bundle. On the other hand it is not too difficult to correct this defect by incorporating an additional projection onto the cotangent space ... 

For Symplectic Euler with constraints, we view Qh as a mapping of the co-tangent bundle and write Q,P) = Qh(s,p) where... 

Lemma 1.3.1. The level surface M23 is diffeomorpbic to the (co)tangent bundle of a two-dimensional sphere. 

By the geodesic flow of the Riemannian manifold M with the metric ds = Y Gijdqidqj we mean a Lagrangian system in the tangent bundle TM with the Langrange function L — Identifying TM with T M by means of the... 

A prominent (and original) instance of vector bundles is that of the so-called tangent bundles. Intuitively, for a manifold M C R of dimension n, one takes A C M X R to be the totality of all tangent spaces to points in M. We refer to [MSta74] for a precise and intrinsic (i.e., independent of the particular embedding of M) definition of the tangent bundle. 

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