Pushforward (differential)

In differential geometry, pushforward is a linear approximation of smooth maps (formulating manifold) on tangent spaces. Suppose that \varphi\colon M\to N is a smooth map between smooth manifolds; then the differential of \varphi at a point x , denoted \mathrm d\varphi_x , is, in some sense, the best linear approximation of \varphi near x . It can be viewed as a generalization of the total derivative of ordinary calculus. Explicitly, the differential is a linear map from the tangent space of M at x to the tangent space of N at \varphi(x) , \mathrm d\varphi_x\colon T_xM \to T_{\varphi(x)}N . Hence it can be used to push tangent vectors on M forward to tangent vectors on N . The differential of a map \varphi is also called, by various authors, the derivative or total derivative of \varphi .

In differential geometry, pushforward is a linear approximation of smooth maps (formulating manifold) on tangent spaces. Suppose that \varphi\colon M\to N is a smooth map between smooth manifolds; then the differential of \varphi at a point x , denoted \mathrm d\varphi_x , is, in some sense, the best linear approximation of \varphi near x . It can be viewed as a generalization of the total derivative of ordinary calculus. Explicitly, the differential is a linear map from the tangent space of M at x to the tangent space of N at \varphi(x) , \mathrm d\varphi_x\colon T_xM \to T_{\varphi(x)}N . Hence it can be used to push tangent vectors on M forward to tangent vectors on N . The differential of a map \varphi is also called, by various authors, the derivative or total derivative of \varphi . If tangent vectors are defined as equivalence classes of the curves \gamma for which \gamma(0) = x, then the differential is given by d\varphi_x(\gamma'(0)) = (\varphi \circ \gamma)'(0). Here, \gamma is a curve in M with \gamma(0) = x, and \gamma'(0) is the tangent vector to the curve \gamma at 0. In other words, the pushforward of the tangent vector to the curve \gamma at 0 is the tangent vector to the curve \varphi \circ \gamma at 0. Alternatively, if tangent vectors are defined as derivations acting on smooth real-valued functions, then the differential is given by d\varphi_x(X)(f) = X(f \circ \varphi), for an arbitrary function f \in C^\infty(N) and an arbitrary derivation X \in T_xM at point x \in M (a derivation is defined as a linear map X \colon C^\infty(M) \to \R that satisfies the Leibniz rule, see: definition of tangent space via derivations). By definition, the pushforward of X is in T_{\varphi(x)}N and therefore itself is a derivation, d\varphi_x(X) \colon C^\infty(N) \to \R . After choosing two charts around x and around \varphi(x), \varphi is locally determined by a smooth map \widehat{\varphi} \colon U \to V between open sets of \R^m and \R^n , and d\varphi_x\left(\frac{\partial}{\partial u^a}\right) = \frac{\partial{\widehat{\varphi}}^b}{\partial u^a} \frac{\partial}{\partial v^b}, in the Einstein summation notation, where the partial derivatives are evaluated at the point in U corresponding to x in the given chart. Extending by linearity gives the following matrix \left(d\varphi_x\right)_a^{\;b} = \frac{\partial{\widehat{\varphi}}^b}{\partial u^a}. Thus the differential is a linear transformation, between tangent spaces, associated to the smooth map \varphi at each point. Therefore, in some chosen local coordinates, it is represented by the Jacobian matrix of the corresponding smooth map from \R^m to \R^n . In general, the differential need not be invertible. However, if \varphi is a local diffeomorphism, then d\varphi_x is invertible, and the inverse gives the pullback of T_{\varphi(x)} N. The differential is frequently expressed using a variety of other notations such as D\varphi_x,\left(\varphi_*\right)_x, \varphi'(x),T_x\varphi. It follows from the definition that the differential of a composite is the composite of the differentials (i.e., functorial behaviour). This is the chain rule for smooth maps. Also, the differential of a local diffeomorphism is a linear isomorphism of tangent spaces. Y_y = \varphi_*\left(X_{\varphi^{-1}(y)}\right). A more general situation arises when φ is surjective (for example the bundle projection of a fiber bundle). Then a vector field X on M is said to be projectable if for all y in N, dφ x (X x ) is independent of the choice of x in φ −1 ({y}). This is precisely the condition that guarantees that a pushforward of X, as a vector field on N, is well defined. Given a Lie group G , we can use the multiplication map m(-,-) : G\times G \to G to get left multiplication L_g = m(g,-) and right multiplication R_g = m(-,g) maps G \to G . These maps can be used to construct left or right invariant vector fields on G from its tangent space at the origin \mathfrak{g} = T_e G (which is its associated Lie algebra). For example, given X \in \mathfrak{g} we get an associated vector field \mathfrak{X} on G defined by \mathfrak{X}_g = (L_g)_*(X) \in T_g G for every g \in G . This can be readily computed using the curves definition of pushforward maps. In the following, we suppose we are working with a matrix Lie group. Then, if we have a curve \gamma: (-1,1) \to G where \gamma(0) = e \, , \quad \gamma'(0) = X we get \begin{align} (L_g)_*(X) &= (L_g\circ \gamma)'(0) \\ &= (g\cdot \gamma(t))'(0) \\ &= \frac{dg}{dt}\gamma(0) + g\cdot \frac{d\gamma}{dt} (0) \\ &= g \cdot \gamma'(0) \end{align} since L_g is constant with respect to \gamma . This implies we can interpret the tangent spaces T_g G as T_g G = g\cdot T_e G = g\cdot \mathfrak{g} . H = \left\{ \begin{bmatrix} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{bmatrix} : a,b,c \in \mathbb{R} \right\} it has Lie algebra given by the set of matrices \mathfrak{h} = \left\{ \begin{bmatrix} 0 & a & b \\ 0 & 0 & c \\ 0 & 0 & 0 \end{bmatrix} : a,b,c \in \mathbb{R} \right\} since we can find a path \gamma:(-1,1) \to H giving any real number in one of the upper matrix entries with i (i-th row and j-th column). Then, for g = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 0 & 0 & 1 \end{bmatrix} we have T_gH = g\cdot \mathfrak{h} = \left\{ \begin{bmatrix} 0 & a & b + 2c \\ 0 & 0 & c \\ 0 & 0 & 0 \end{bmatrix} : a,b,c \in \mathbb{R} \right\} which is equal to the original set of matrices. This is not always the case, for example, in the group G = \left\{ \begin{bmatrix} a & b \\ 0 & 1/a \end{bmatrix} : a,b \in \mathbb{R}, a \neq 0 \right\} we have its Lie algebra as the set of matrices \mathfrak{g} = \left\{ \begin{bmatrix} a & b \\ 0 & -a \end{bmatrix} : a,b \in \mathbb{R} \right\} hence for some matrix g = \begin{bmatrix} 2 & 3 \\ 0 & 1/2 \end{bmatrix} we have T_gG = \left\{ \begin{bmatrix} 2a & 2b - 3a \\ 0 & -a/2 \end{bmatrix} : a,b\in \mathbb{R} \right\} which is not the same set of matrices.

See also

Pullback (differential geometry) Flow-based generative model

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