I am trying to understand the asymptotic heat trace expansion \begin{equation} \text{Tr}(e^{-t\triangle_g}) \backsim \sum_{k \geq 0} t^{k - \frac{n}{2}}c_{2k} \quad (t \to 0^+) \end{equation} that is associated with the Laplace - Beltrami operator \begin{equation} \triangle_g = \frac{1}{\sqrt{\text{det}(g)}} \sum^n_{i,j = 1} \frac{\partial}{\partial x_i} g^{ij} \sqrt{\text{det}(g)} \frac{\partial}{\partial x_j} \end{equation} where $g$ denotes the metric on the n - dimensional Riemannian Manifold $(M,g)$ that we consider.
Now, I understand that the first invariant $c_0$ is equal to the Volume of $M$ and the next coefficient is associated with the Scalar curvature (is that correct?). If I take $M$ to be 1-dimensional compact and boundaryless, then this means the second coefficient should evaluate to zero, is that correct ?
Many thanks for your help!