In differential geometry, pushforward is a linear approximation of smooth maps (formulating manifold) on tangent spaces. Suppose that φ : M → N {\displaystyle...

Pushforward (differential), the differential of a smooth map between manifolds, and the "pushforward" operations it defines Pushforward (homology), the...

\right)_{X}\left(\phi ^{*}s\right)=\phi ^{*}\left(\nabla _{d\phi (X)}s\right).} Pushforward (differential) Pullback bundle Pullback (category theory) Jost, Jürgen (2002)...

field Tensor field Differential form Exterior derivative Lie derivative pullback (differential geometry) pushforward (differential) jet (mathematics)...

N, the pushforward of a vector field does not always exist. By contrast, it is always possible to pull back a differential form. A differential form on...

interpreted as infinitesimals Differential topology Differential (pushforward) The total derivative of a map between manifolds. Differential exponent, an exponent...

the differential or pushforward refers to the derivative of a map between smooth manifolds and the pushforward operations it defines. The differential is...

_{*}W_{v}=v\,} at each point v ∈ TM; here π∗ : TTM → TM denotes the pushforward (differential) along the projection π : TM → M associated to the tangent bundle...

F:M\,\longrightarrow \,N} be a smooth surjection such that the pushforward (differential) of F {\displaystyle \scriptstyle F} is surjective almost everywhere...

of df must be a velocity. If one treats the differential in this manner, then it is known as the pushforward since it "pushes" velocities from a source...

sensitivity and statistical diagnostics. Center manifold Hessian matrix Pushforward (differential) Differentiability at x implies, but is not implied by, the existence...

spaces: for F : M → N {\displaystyle F:M\to N} , at each point the pushforward (or differential) maps tangent vectors at p {\displaystyle p} to tangent vectors...

*:A^{k}\rightarrow A^{n-k}} . as a unary operator, written as a subscript The pushforward (differential) of a smooth map f between two smooth manifolds, denoted f∗. And...

GL(V) is a group representation for some vector space V, then its pushforward (differential) at the identity, or Lie map, π : g → End V {\displaystyle \pi...

{\displaystyle \pi :TM\rightarrow M} is the canonical projection. Pushforward (differential) Unit tangent bundle Cotangent bundle Frame bundle Musical isomorphism...

{\displaystyle T_{F(p_{0})}N,} with a linear map between them given by the pushforward (differential): F ∗ , p : T p 0 M → T F ( p 0 ) N . {\displaystyle F_{*,p}:T_{p_{0}}M\to...

In differential geometry, the cotangent space is a vector space associated with a point x {\displaystyle x} on a smooth (or differentiable) manifold M...

this topology may be trivial without the diffeology being trivial). The pushforward diffeology of a diffeological space X {\displaystyle X} by a function...

)}gd\mu =\int _{\Omega }g\circ TdT^{*}\mu } . Pushforward measure and transformation formula The pushforward measure in terms of a transformation T {\displaystyle...

In differential geometry, the Lie derivative (/liː/ LEE), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including...

H^{*}(E).\,} In the case of a fiber bundle, one can also define a pushforward map π ∗ {\displaystyle \pi _{\ast }} π ∗ : H ∗ ( E ) ⟶ H ∗ − k ( M )...

D-modules on different algebraic varieties are connected by pullback and pushforward functors comparable to the ones for coherent sheaves. For a map f: X...

is a point of U0 ⊂ S, then a vector field may be represented by the pushforward of a vector field v0 on R2 by φ 0 {\displaystyle \varphi _{0}} : where...

so that although the notion of a 'pushforward of a bundle' is defined in some contexts (for example, the pushforward by a diffeomorphism), in general it...

tangent vector will agree with the directional derivative. The differential or pushforward of a map between manifolds is the induced map between tangent...

but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition with a function probably provides the most elementary...

differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator...

or to a pair of maps to the ambient manifold, by asking whether the pushforwards of the tangent spaces along the preimage of points of intersection of...

of the tangent space TgG at each g ∈ G into TeG. It is given as the pushforward of a vector in TgG along the left-translation in the group: ω ( v ) =...

orientation-preserving diffeomorphisms of S1, Diff(S1), acts on Fλ(S1) via pushforwards. If f is an element of Diff(S1) then consider the mapping f → S ( f −...