# 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