I have a Riemannian manifold $\mathcal{M}$ and a closed curve in such manifold (say $\mathcal{C}$) what I want to workout is the Laplace Beltrami operator of $\mathcal{C}$ (with the metric restricted to $\mathcal{C}$). Also assume $\mathcal{C} = \Gamma((a,b))$.
My attempt is the following. Since on $\mathcal{M}$ we have
$$ \Delta_{\mathcal{M}} = \frac{1}{\sqrt{|g|}} \sum_{i,j} \partial_i \sqrt{|g|} g^{ij} \partial_j f $$
My thought is to observe that on $\mathcal{C}$ we have
$$ \Delta_{\mathcal{C}} f = \frac{1}{\sqrt{h}} \frac{d}{dt} \left( \frac{\sqrt{h}}{h} \frac{d}{dt}f \right) = \frac{1}{\sqrt{h}} \frac{d}{dt} \left( \frac{1}{\sqrt{h}} \frac{d}{dt}f \right) = \frac{1}{\sqrt{h}} \left(- \frac{1}{2 h\sqrt{h}} \frac{dh}{dt} \frac{df}{dt} + \frac{1}{\sqrt{h}} \frac{d^2f}{dt^2}\right) = - \frac{1}{2h^2} \frac{dh}{dt} \frac{df}{dt} + \frac{1}{h} \frac{d^2f}{dt^2} $$
Where $h$ is the Riemannian Metric of $\mathcal{C}$, which is a positive scalar function since the dimension of the $\mathcal{C}$ is 1. Since I am restricting the metric from $\mathcal{M}$ to $\mathcal{C}$ to find $h$ it can be defined from
$$ h(t) = \left\langle \frac{d}{dt} , \frac{d}{dt} \right\rangle_{\mathcal{C}}(t) = \left\langle \sum_i a_i(\Gamma(t)) \frac{\partial}{\partial x_i}, \sum_j a_j (\Gamma(t)) \frac{\partial}{\partial x_j} \right\rangle_{\mathcal{M}}(\Gamma(t)) = \sum_{i,j} a_i(\Gamma(t)) a_j(\Gamma(t)) \left\langle \frac{\partial}{\partial x_i}, \frac{\partial}{\partial x_j} \right\rangle_{\mathcal{M}}(\Gamma(t)). $$
Without loss of generality I can assume the Rienmannian metric is orthonormal and therefore I can write
$$ \sum_{i,j} a_i(\Gamma(t)) a_j(\Gamma(t)) \left\langle \frac{\partial}{\partial x_i}, \frac{\partial}{\partial x_j} \right\rangle_{\mathcal{M}}(\Gamma(t)) = \sum_i a_i(\Gamma(t))^2. $$
By simple calculations it can be showong that each $a_i(\Gamma(t)) = \left. \frac{d}{dt} \right|_t (x^i \circ \Gamma) = \frac{d x^i}{dt}$, so I can write
$$ \Delta_{\mathcal{C}} f = -\frac{1}{\sum_i \left( \frac{dx^i}{dt} \right)^2} \sum_j \frac{dx^j}{dt} \frac{d^2x^j}{dt^2} \frac{df}{dt} + \frac{1}{\sqrt{\sum_i \left( \frac{dx^i}{dt} \right)^2}} \frac{d^2 f}{dt^2} $$
Are these calculations correct? I think they're cause in the case of $h = 1$ I get the second order differential operator.
Can anyone confirm?