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In Mathematics , a pullback can be defined in Several Different Contexts . This article focuses primarily on the pullback of Tensors on Differentiable Manifold s.

  • ''. More generally, any Covariant tensor pulls back under ''f---''.


When the map ''f'' is a Diffeomorphism of a manifold to itself, then the pullback, together with the Pushforward , describe the transformation properties of the manifold under a Change Of Coordinates . Using traditional language, these describe the transformation properties of Contravariant and Covariant tensors.


PULLBACK ON TENSORS

  • T on ''V'' can be defined. That is, given a tensor

    • :T:W imes W imes \cdots imes W ightarrow \mathbb{R}

and a set of vectors
:(v_1,v_2,\ldots,v_n) \in V imes V imes \cdots imes V
one then defines the pullback as
  • T)(v_1,v_2,\ldots,v_n) = T(f(v_1), f(v_2), \ldots ,f(v_n)).


  • T is again a tensor, so that f^--- is in fact a mapping from tensors on ''W'' to tensors on ''V''. As a special case, note that if ''T'' is a (0,1)-tensor, so that T\in W^---, the Dual Space of ''W'', then f^---T\in V^---, and so the pullback acts in the reversed direction:


  • :W^--- ightarrow V^---.


  • :V^--- ightarrow W^---; we make use of the identity (f^{-1})^---=(f^---)^{-1}. One then defines


  • T)(v^---_1,v^---_2,\ldots,v^---_n) =

  • (v^---_1), (f^{-1})^---(v^---_2), \ldots ,(f^{-1})^---(v^---_n))


Thus we've shown that if a linear transformation is invertible, it can be used to define the pullback on general tensors of mixed rank (m,n). Perhaps the easiest way to visualize and understand the above is to keep firmly in mind that ''f'' is nothing more than a Matrix , so that ''f''(''v'') is just the multiplication of a vector by a matrix. Similarly, the dual space should be visualized as nothing more than a Dot Product .


PULLBACK OF (CO)TANGENT BUNDLES

The pullback of Smooth Map ''f'' : ''M'' → ''N''
between :



  • ''M'' and ''T''---''N'' are the Cotangent Bundle s of ''M'' and ''N'' respectively, and π''M'' and π''N'' are the natural projections. Perhaps the easiest way to understand the pullback is in terms of the Pushforward of ''f''. Picking a point p\in M, the pushforward at ''p'' is a linear map between the tangent spaces


  • (p) \in L(T_pM,T_{f(p)}N)


where L(V,W) being the set of linear mappings from the vector space V = T_pM to the vector space W = T_{f(p)}N. The Cotangent Space is Dual to the tangent space, and maps on the dual space act as the Transpose . That is, consider two ordinary vectors ''v'' and ''w'', and a Matrix ''A''. The Dot Product obeys the identity w\cdot Av = (A^Tw)\cdot v. Thus, if we take

  • (p)\in L(T_pM,T_{f(p)}N)


then the transpose is going to act on the 1-form s:

  • (p)" class="copylinks" target="_blank">{Link without Title} ^T\in L(T^---_{f(p)}N,T^---_pM ).


We use the transpose to map the cotangent spaces. For each point in the manifold, the pullback is defined as the matrix transpose of the pushforward; that is,


Note that this mapping is in a certain sense going in the "backwards" direction, that is,

  • :T^---N ightarrow T^---M.



PULLBACK ON TENSOR BUNDLES

  • =T^---_pM at point ''p'' in ''M'', one defines the tensor space at point ''p'' as the n-fold tensor product


  • \otimes V^--- \otimes ... \otimes V^---.


  • =T^---_{f(p)}N. This definition applies as well to the Exterior Bundle s Λ''k''''T''---''N'' and Λ''k''''T''---''M'', are strict subspaces of the general tensor bundles, closed under the Exterior Algebra . The pullback operation commutes with the Exterior Algebra , and so the pullback of an Alternating Form is again an alternating form. That is, the pullback of a Differential Form on ''N'' is a differential form on ''M''. Symbolically, we write


  • (\alpha \wedge \beta)=f^---\alpha \wedge f^---\beta


for α and β in Λ(M). Similarly, the pullback is natural with respect to derivations:

  • (d\omega) = d(f^---\omega)


for ω in Λ(M).


PULLBACK OF DIFFEOMORPHISMS

When the map ''f'' between manifolds is a Diffeomorphism , that is, it is both smooth and invertible, then the pullback can be defined for the Tangent Space as well as for the Cotangent Space , and thus, by extension, for an arbitrary mixed tensor bundle on the manifold. The matrix

  • (p)\in GL(T_pM,T_{f(p)}N)


can be inverted to define

  • (p)" class="copylinks" target="_blank">{Link without Title} ^{-1} \in GL(T_{f(p)}N, T_pM)


and thus one has, at each point ''p'', that the pushforward is the inverse of the pullback, now acting on the tangent space (instead of the cotangent space):


so that

  • :TN ightarrow TM.


A general mixed tensor will then transform as a mixture of transposes and inverses, depending on whether the indices are contra- or co-variant. When ''M'' = ''N'', then the pullback and the Pushforward describe the transformation properties of a Tensor on the manifold ''M''. In traditional terms, the pullback describes the transformation properties of the Covariant indices of a Tensor ; by contrast, the transformation of the Contravariant indices is given by a Pushforward .


SEE ALSO



REFERENCES

  • Jurgen Jost, ''Riemannian Geometry and Geometric Analysis'', (2002) Springer-Verlag, Berlin ISBN 3-540-4267-2 ''See sections 1.5 and 1.6''.

  • Ralph Abraham and Jarrold E. Marsden, ''Foundations of Mechanics'', (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X ''See section 1.7 and 2.3''.