multilineal lección 6


tensor fields on a surface and its covariant derivatives

the collection T\Sigma of all the tangent spaces T_p\Sigma is called the tangent bundle of the surface, i.e.

T\Sigma=\bigsqcup_pT_p\Sigma

A vector field in a surface is a mapping X:\Sigma\to T\Sigma with the condition p\mapsto X\in T_p\Sigma and since \partial_1,\partial_2 span T_p\Sigma then

X=X^s\partial_s

This construction determines a contravariant tensor field of rank one, which is taken as the base to ask how other tensor fields -of any rank and any variance- vary.

A contravariant tensor field of rank two in a surface is a mapping \Sigma\to T\Sigma\otimes T\Sigma which attach to each point of the surface a tensor of rank two

p\mapsto B=B^{\mu\nu}\partial_{\mu}\otimes\partial_{\nu}\in T_p\Sigma\otimes T_p\Sigma

where the scalars B^{\mu\nu} are function around the point p, so if one is interested in the concept of  how B varies in the direction X then one is compelled to attach a meaning to \nabla_XB for which the most natural way is by means of the Leibniz’s rule:

\nabla_XB=\nabla_X(B^{\mu\nu}\partial_{\mu}\otimes\partial_{\nu})

=X(B^{\mu\nu})\partial_{\mu}\otimes\partial_{\nu}+B^{\mu\nu}\nabla_X(\partial_{\mu}\otimes\partial_{\nu})

where you have to remember that the directional derivative of a scalar is Xf=\langle X,{\rm{grad}}f\rangle and also that we can abbreviate \partial_kf=f_{,k} as well.

 To simplify the grasping of the idea let us illustrate first by calculating the components of

\nabla_{\partial_k}B

\nabla_{\partial_k}B=\nabla_{\partial_k}(B^{\mu\nu}\partial_{\mu}\otimes\partial_{\nu})

\nabla_{\partial_k}B=(B^{\mu\nu})_{,k}\partial_{\mu}\otimes\partial_{\nu}+B^{\mu\nu}(\nabla_{\partial_k}\partial_{\mu})\otimes\partial_{\nu}+B^{\mu\nu}\partial_{\mu}\otimes(\nabla_{\partial_k}\partial_{\nu})

\nabla_{\partial_k}B=(B^{\mu\nu})_{,k}\partial_{\mu}\otimes\partial_{\nu}+B^{\mu\nu}({\Gamma^s}_{k\mu}\partial_s)\otimes\partial_{\nu}+B^{\mu\nu}\partial_{\mu}\otimes({\Gamma^s}_{k\nu}\partial_s)

\nabla_{\partial_k}B=[(B^{\mu\nu})_{,k}+B^{s\nu}{\Gamma^{\mu}}_{ks}+B^{\mu s}{\Gamma^{\nu}}_{ks}]\partial_{\mu}\otimes\partial_{\nu}

With this technique it is really easy to prove that for the metric tensor g=g^{ij}\partial_i\otimes\partial_j we have:

\nabla_{\partial_k}g=0

that is: the metric tensor is covariantly constant. In the process you gonna need to use the formula

{\Gamma^i}_{jk}=\frac{1}{2}g^{is}[g_{sk,j}+g_{js,k}-g_{jk,s}]

Another type of phenomena that you’ll gonna face is the problem of determine how is the behaviour of tensor quantities (of any rank and any variaance) defined in a surface it is with respect to change of parameterizations or change of coordiantes.

A surface can be GPS-ed with many diverse parameterization, zum beispiel for the two-sphere S^2 in \mathbb{R}^3, we show three:

  • \left(\!\!\begin{array}{c}v\\w\end{array}\!\!\right)\stackrel{\phi_1}\to\left(\!\!\begin{array}{c}v\\\\w\\\\\sqrt{1-v^2-w^2}\end{array}\!\!\right)
  • .
  • \left(\!\!\begin{array}{c}v\\w\end{array}\!\!\right)\stackrel{\phi_2}\to\left(\!\!\begin{array}{c}\cos v\cos w\\\\\cos v\sin w\\\\\sin v\end{array}\!\!\right)
  • .
  • \left(\!\!\begin{array}{c}v\\w\end{array}\!\!\right)\stackrel{\phi_3}\to\left(\begin{array}{c}\frac{2v}{1+v^2+w^2}\\\\\frac{2w}{1+v^2+w^2}\\\\\frac{1-v^2-w^2}{1+v^2+w^2}\end{array}\right)

for these three maps one must choose a suitable domain for v,w to ensure injectivity. The maps  \lambda_{ij}={\phi_j}^{-1}\cdot{\phi_i} are called change of coordinates for the surface. Note that for any two of them we have {\phi_j}\cdot\lambda_{ij}=\phi_i.

With a device like that it is possible to describe how the tangent coordinated basis change when different parameterization are taken…

A rank 2 tensor can constructed by asking how are the components of

\nabla_{\partial/\partial x^k}X

\nabla_{\partial_k}X

\nabla_{\partial_{ x^k}}X

\nabla_{k}X

the answer is by doing Leibniz in \nabla_{\partial_k}(X^s\partial_s). So…

\nabla_{\partial_k}(X^s\partial_s)=(\nabla_{\partial_k}X^s)\partial_s+X^s(\nabla_{\partial_k}\partial_s)

=(\partial_kX^s)\partial_s+X^s{\Gamma^t}_{ks}\partial_t=({X^s}_{,k}+X^t{\Gamma^s}_{kt})\partial_s

so

{X^s}_{;k}={X^s}_{,k}+X^t{\Gamma^s}_{kt}

are the components of this tensor.

This can be considered as a measure of what so far are each other:

  • the std-calculus ({\Gamma^i}_{jk}=0) and
  • the with-curvature-calculus {\Gamma^i}_{jk}\neq0

 The {\Gamma^i}_{jk} they code or detect curvature

 … continuará

but meanwhile, you would like to peer (to constrast and extra grasp) on Octavian lessons at manifolds 1 and manifolds 2

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