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The tangent space and the cotangent space at a point are both real vector spaces of the same dimension and therefore isomorphic to each other via many possible isomorphisms. The introduction of a Riemannian metric or a symplectic form gives rise to a natural isomorphism between the tangent space and the cotangent space at a point, associating ...
Then there is an associated linear map from the space of 1-forms on (the linear space of sections of the cotangent bundle) to the space of 1-forms on . This linear map is known as the pullback (by ϕ {\displaystyle \phi } ), and is frequently denoted by ϕ ∗ {\displaystyle \phi ^{*}} .
For example, this is a way to describe the phase space of a pendulum. The state of the pendulum is determined by its position (an angle) and its momentum (or equivalently, its velocity, since its mass is constant). The entire state space looks like a cylinder, which is the cotangent bundle of the circle.
denotes the cotangent space of at the point . Sections of the ... If : is a smooth map, then the pull-back of a -form () is defined such that for any ...
Tangent space; Tangent bundle; Cotangent space; Cotangent bundle; Tensor; Tensor bundle; Vector field; Tensor field; Differential form; Exterior derivative; Lie derivative; pullback (differential geometry) pushforward (differential) jet (mathematics) Contact (mathematics) jet bundle; Frobenius theorem (differential topology) Integral curve
By contrast, it is always possible to pull back a differential form. A differential form on N may be viewed as a linear functional on each tangent space. Precomposing this functional with the differential df : TM → TN defines a linear functional on each tangent space of M and therefore a differential form on M. The existence of pullbacks is ...
Just as a complex structure on a vector space V allows a decomposition of V C into V + and V − (the eigenspaces of J corresponding to +i and −i, respectively), so an almost complex structure on M allows a decomposition of the complexified tangent bundle TM C (which is the vector bundle of complexified tangent spaces at each point) into TM ...
The pullback diffeology of a diffeological space by a function : is the diffeology on whose plots are maps such that the composition is a plot of . In other words, the pullback diffeology is the smallest diffeology on X {\displaystyle X} making f {\displaystyle f} differentiable.