For Riemannian surfaces in $\mathbb R^n,n\geq 3$, the Laplace-Beltrami operator can be expressed in terms of the Riemannian metric and one can consider evolution equations (e.g. heat diffusion, wave equation) on the surface, where the curvature of the surface is taken into account.
Now, consider a parametrized curve $\gamma: [0,1]\rightarrow \mathbb{R}^2$. I would like to consider PDEs along $\gamma$, e.g. heat diffusion of some function $u(t,x), x\in \gamma([0,1]), t\geq 0$.
How can one account for the curvature of $\gamma$? Or formulated differently, how can one express the Laplacian in terms of $\gamma$ and its derivatives?