Let $M^{k}$ a submanifold, $h:U\to M^{k}$ a chart, and $\gamma:[a,b]\to h(U)\subset M^{k}$ a curve in $M^{k}$. Represent the curve in coordinates $(h,U)$ as $h^{-1}\circ\gamma(t)=(y^{1}(t),...,y^{k}(t))$. Show that the lenght of the curve can be calculate in function of coefficients of Riemannian metric $g_{ij}$ in the chart $(h,U)$ by the formula $$l(\gamma)=\int_{a}^{b}{\sum_{i,j=1}^{k}{g_{ij}(\gamma(t))\dfrac{dy^{i}(t)}{dt}\dfrac{dy^{j}(t)}{dt}dt}}$$

Hi, I have this exercises and yet I cannot conclude anything. The Lenght of the curve is the expresion $$l(\gamma)=\int_{a}^{b}{g_{\gamma(t)}(\dot{\gamma}(t)),\dot{\gamma}(t))^{\frac{1}{2}}dt}$$ Any hint or idea, thanks!


Define a metric on $U$ where $h: U\rightarrow M$ : $$ g_{ij}:= h^\ast g (e_i,e_j) $$

If $\gamma : [a,b] \rightarrow M$ is a curve, then $$h^{-1}\circ \gamma (t)= (y^1(t),\cdots, y^n(t)) $$ is a curve in $U$

So its tangent is $$ \frac{d}{dt} h^{-1}\circ \gamma =(\frac{d}{dt} y^1(t),\cdots, \frac{d}{dt}y^n(t)) $$ Hence $$ h^\ast g ( \frac{d}{dt} h^{-1}\circ \gamma , \frac{d}{dt} h^{-1}\circ \gamma ) = \frac{d}{dt}y^i\frac{d}{dt} y^j h^\ast g (e_i,e_j)=\frac{d}{dt}y^i\frac{d}{dt} y^j g_{ij} $$

So $$ L=\int_a^b \sqrt{\frac{d}{dt}y^i\frac{d}{dt} y^j g_{ij} } $$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.