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.


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


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:



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=[(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:


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


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_{ x^k}}X


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





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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