Lagmental Vicfred

Levi--Civita Is the Unique Metric Torsion-Free Connection by Vicfred

tags: mathematics, Riemannian geometry, geodesics, connections

One connection uniquely satisfies compatibility with the metric and symmetry of covariant derivatives. This is a compact note, but the quantifiers and hypotheses stay on the page.

Definitions first

A Riemannian metric \(g_p:T_pM\times T_pM\to\mathbf R\) varies smoothly and assigns lengths and angles. In coordinates it is a positive-definite matrix \((g_{ij})\).

$$ \nabla g=0,\qquad\nabla_XY-\nabla_YX=[X,Y] $$

I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.

$$ 2g(\nabla_XY,Z)=Xg(Y,Z)+Yg(Z,X)-Zg(X,Y)-g(X,[Y,Z])+g(Y,[Z,X])+g(Z,[X,Y]) $$

A small case in full

A worked instance is useful here because it exposes every index that the compressed statement hides.

$$ \Gamma^k_{ij}=\frac12g^{k\ell}\left(\frac{\partial g_{j\ell}}{\partial x^i}+\frac{\partial g_{i\ell}}{\partial x^j}-\frac{\partial g_{ij}}{\partial x^\ell}\right) $$

The reusable statement

The abstraction earns its keep by explaining why the same computation reappears. The notation compresses repeated reasoning without erasing the hypothesis that licenses it.

$$ \begin{aligned} \mathsf{D}\;&:\quad \nabla g=0,\qquad\nabla_XY-\nabla_YX=[X,Y],\\[5pt] \mathsf{C}\;&:\quad 2g(\nabla_XY,Z)=Xg(Y,Z)+Yg(Z,X)-Zg(X,Y)-g(X,[Y,Z])+g(Y,[Z,X])+g(Z,[X,Y]). \end{aligned} $$

A nearby false statement

Christoffel symbols depend on coordinates even though the Levi--Civita connection and geodesic equation are intrinsic.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] 2g(\nabla_XY,Z)=Xg(Y,Z)+Yg(Z,X)-Zg(X,Y)-g(X,[Y,Z])+g(Y,[Z,X])+g(Z,[X,Y]) \end{gathered}} $$

The notation is dense, but it is doing honest work: every delimiter records scope and every index records dependence. Removing one should require a mathematical reason.

This article was posted on Thu 06 June 2019. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.