# Scalar curvature of sum of constant curvature metrics

Suppose $$M$$ is a surface and $$g_0$$, $$g_1$$ are two Riemannian metrics having the same constant Gaussian curvature $$K$$. Is there anything that can be said about the Gaussian curvature of $$g_t := (1-t)g_0 + tg_1$$ where $$0 \leq t \leq 1$$?

• I think it was asked before and the answer was "no." Aug 6, 2023 at 22:23
• Any chance you can say a little about how this question arose and whether there is a conclusion you are hoping for? Locally, $g_1 = u^*g_0$ where $u$ is a diffeomorphism. Here's a suggestion: Let $h = g_1-g_0$. The difference between the connections is a tensor involving covariant derivatives of $h$, which in turn can be written in terms of second order covariant derivatives of $u$. The Gauss curvature will have a formula in terms of third order covariant derivatives of $h$. Maybe if you calculate the formulas, you might see something interesting there. Maybe what you're hoping for. Aug 6, 2023 at 23:17

## 1 Answer

Comment only.

Any chance you can say a little about how this question arose and whether there is a conclusion you are hoping for? Locally, $$g_1 = u^*g_0$$ where $$u$$ is a diffeomorphism. Here's a suggestion: Let $$h = g_t-g_0$$. The difference between the connections is a tensor involving covariant derivatives of $$h$$ with respect to the Levi-Civita connection of $$g_0$$, which in turn can be written in terms of second order covariant derivatives of $$u$$. The Gauss curvature of $$g_t$$ will have a formula in terms of the Gauss curvature of $$g_0$$ and third order covariant derivatives of $$u$$. Maybe if you calculate the formulas, you might see something interesting there. Maybe what you're hoping for.