Let $\hat{g}$ and $\bar{g}$ be two smooth Riemannian metrics defined, say, on $\mathbb{R}^n.$ Consider a smooth function $\xi$ that acts as an interpolation function between the two metrics above on an annulus $B\_{2R}(0)\backslash B\_R(0).$


If we define $g=\xi\hat{g} + (1-\xi)\bar{g},$ which is clearly a smooth Riemannian metric, can we find write $Rm(g)$ in terms of $Rm(\hat{g})$ and $Rm(\bar{g})$ plus an error? More importantly, what would be the structure of this error?


I am sure someone did this calculation already, but I cannot find it anywhere. The calculation gets very messy very quickly, so I would appreciate a reference if anyone has one. Thank you.



Curvature tensor of interpolation of two metrics
After some Googling, the trick seems to be to consider the difference between the two metrics $ h = \tilde{g}-\bar{g}.$ Thus, $g= \bar{g} + \xi h.$ Using the notation above, we can write the Christoffel symbols of $g$ schematically as:


$\Gamma(g) = \Gamma(\bar{g}) + \frac{1}{2}g \* \bar{\nabla}(\xi h),$


where $ h = \tilde{g}-\bar{g}.$ Thus,


$Rm(g)= Rm(\bar{g}) + g^{-1} \*[ \bar{\nabla}^2\xi \*h + \bar{\nabla}\xi\*\bar{\nabla}h+ \xi^2\bar{\nabla}^2h] + g^{-2}[\*\bar{\nabla}\xi\*\bar{\nabla}\xi\*h\*h + \bar{\nabla}h\*\bar{\nabla}\xi\*\xi h +\xi^2\*\bar{\nabla}h\*\bar{\nabla}h],$


where $A\*B$ denotes any linear combination of terms involving contractions of the tensor product $A\otimes B.$


The details can be found in this paper by Gursky and Viaclovsky: <https://www.sciencedirect.com/science/article/pii/S0001870815301596?via%3Dihub>.


