Answer to Question #110779 in Differential Geometry | Topology for brighton

Question

1. Find the smooth curve (C1) realizing the shortest distance between two given points in a plan (usual Euclidean R2).

Answer 1

$\frac{d^2x^{\lambda}}{dt^2}+\Gamma^{\lambda}_{\mu\nu}\frac{dx^{\mu}}{dt}\frac{dx^{\nu}}{dt}=0$ is geodesic equation.

For $\mathbb R^2$ we have $g_{11}=g_{22}=1$, $g_{12}=g_{21}=0$. So since $\Gamma^{\lambda}_{\mu\nu}=\frac{1}{2}g^{\lambda m}\left(\frac{\partial g_{m\mu}}{\partial x^\nu}+\frac{\partial g_{m\nu}}{\partial x^\mu}-\frac{\partial g_{\mu\nu}}{\partial x^m}\right)$, we have $\Gamma^{\lambda}_{\mu\nu}=0$ for every $(\lambda,\mu,\nu)\in\{1,2\}^3$.

Then geodesic equation for $\mathbb R^2$ is $\frac{d^2x^1}{dt^2}=0, \frac{d^2x^2}{dt^2}=0$. We have $x^1=At+B, x^2=Ct+D$, that is $ACt=C(x^1-B)=A(x^2-D)$, so $Ax^2+Ex^1+F=0$, where $E=-C, F=-AD+BC$.

It is equation of straight line.

Answer: straight line between two given points.