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The differential and pullback

If we consider a general mapping between manifolds Φ:MmNn, we can choose charts αM:MRm and αN:NRn, with coordinate functions xμ and yν, so that the mapping αNΦ:MRn can be represented by n functions Φν:MR. This allows us to write down an expression for the induced tangent mapping or differential (aka pushforward, derivative) dΦ:TMTN (also denoted TΦ or Φ or sometimes simply Φ if it is clear the argument is a tangent vector). For a tangent vector v=vμ/xμ at a point pM we define

dΦ(v)|pvμΦλxμyλ|Φ(p).

This definition can be shown to be coordinate-independent and to follow our intuitive expectation that mapped tangent vectors stay tangent to mapped curves. If M=N and Φ is the identity, dΦ is just the vector component transformation from the previous section on tangent vectors. The matrix JΦ(x)Φν/xμ is called the Jacobian matrix (AKA Jacobian). For the parametrized curve C:RNn, we define the tangent to the curve at tR to be

˙C(t)dC(x)|t=Cλxyλ|C(t),

which is also denoted dC(t)/dt and coincides with the Euclidean tangent to a curve if N=Rn.

If Φ is a diffeomorphism, dΦ is an isomorphism between the tangent spaces at every point in M. The inverse function theorem says that the converse is true locally: if dΦp is an isomorphism at pM, then Φ is locally a diffeomorphism. In particular, this means that if in some coordinates the Jacobian is nonsingular, then αNΦα1M represents a locally valid coordinate transformation and Φν=yν.

A mapping between manifolds Φ:MmNn also can be used to naturally define the pullback of a form Φ:ΛkNΛkM by Φφ(v1,,vk)=φ(dΦ(v1),,dΦ(vk)), where the name indicates that a form on N can be “pulled back” to M using Φ. Note that the composition of pullbacks is then ΨΦφ=(ΦΨ)φ.

39.pullback-v2

Note that for a mapping f:MR, we have df:TMTRR, so that df(v)=vμf/xμ=v(f), the directional derivative of f. Let us apply this to the coordinate function x1:MR. Then we have dx1(v)=vμx1/xμ=v1, so that in particular dxv(/xμ)=δvμ, i.e. dxμ is in fact the dual frame to /xμ. Thus in a given coordinate system, we can write a general tensor of type (m,n) as

T=Tμ1μmν1νnxμ1xμmdxν1dxνn.

In particular, the metric tensor is often written

ds2g=gμνdxμdxν,

where the Einstein summation convention is used and the tensor symbol omitted. A general k-form φΛkM can then be written as

φ=μ1<<μkφμ1μkdxμ1dxμk.

From either the tangent mapping definition or the behavior of the exterior product under a change of basis, we see that under a change of coordinates we have

dyμ1dyμk=det(yνxμ)dxμ1dxμk.

This is the familiar Jacobian determinant (like the Jacobian matrix, also often called the Jacobian) that appears in the change of coordinates rule for integrals from calculus, and explains the name of the volume form as defined previously in terms of the exterior product.

In summary, the differential d has a single definition, but is used in several different settings that are not related in an immediately obvious way.

ConstructArgumentOther namesOther symbols
dΦ:TMTNΦ:MNTangent mappingTΦ, Φ, Φ
df:TMRf:MRDirectional derivativev(f), dvf, vf
dxμ:TMRxμ:MRDual frame to /xμβμ

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