I believe my question to be most likely trivial for a differential geometer, however, I have been having problems with this — partly, I suspect, due to the different nomenclature used by theoretical physicists (like me) and mathematicians.
Let there be a one-dimensional curve in a two-dimensional space. (The 2D space is not embedded in any higher dimensions.) Now, I know a curve has no intrinsic curvature, however, it does have an extrinsic notion of curvature in the 2D space: for example, the geodesic curvature $k_g$. I also know that, in general, given an ($n$-1)-dimensional hypersurface in an $n$-dimensional manifold, one can define the extrinsic curvature tensor $K_{\mu \nu}$ (second fundamental form) of the hypersurface. My question is, for a 1D hypersurface (i.e., a curve) in a 2D manifold, is the trace of its extrinsic curvature tensor equal to its geodesic curvature? I.e., is $K=h^{\mu \nu}K_{\mu \nu}$ equal to $k_g$ for the curve, with $h^{\mu \nu}$ the raised-indices metric tensor of the curve?
The motivation for this question is the following: I am using the Gauss-Bonnet theorem in my work and I know of two different but seemingly equivalent formulations for a two-dimensional manifold $M$ with one-dimensional boundary $\partial M$:
$\hspace{0.7cm} (1) \hspace{0.7cm} \chi (M) = \frac{1}{4\pi} \int_M \sqrt{g} R d^{2} x + \frac{1}{2\pi} \int_{\partial M} \sqrt{h} K dx$ ; $\hspace{0.7cm} (2) \hspace{0.7cm} \chi (M) = \frac{1}{4\pi} \int_M \sqrt{g} R d^{2} x + \frac{1}{2\pi} \int_{\partial M} \sqrt{h} k_g dx$
where $\chi (M)$ is the Euler characteristic of the manifold, $R$ is the Ricci scalar (scalar curvature) of the manifold, $g$ is the determinant of the metric tensor on the manifold, and $h$ is the determinant of the metric tensor on the boundary. Clearly, by comparing the boundary terms of formulae (1) and (2), it is implied that $K=k_g$ for a (closed) curve in a 2D space, but I have been unable to prove this so far.
Any help would be much appreciated! 🙂
Best Answer
I think this is a notational confusion. There is really no trace to take when you have a $2$-tensor on a $1$-dimensional manifold (i.e., in coordinates, you have a $1\times 1$ matrix).
But let's look at the definitions. If $e_1,e_2$ is an adapted orthonormal frame, so $e_1$ is tangent to the curve and $e_2$ normal, then the geodesic curvature is $k_g = e_1'\cdot e_2$ (where $'$ denotes differentiation with respect to arclength of the curve). The second fundamental form is $f ds\otimes ds$, with $f=-e_2'\cdot e_1=e_1'\cdot e_2$. So, yes, $f=k_g$, assuming the correct choice of normal vector.