Timeline for Is the space of Levi-Civita connections convex

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Nov 24, 2016 at 14:19	comment	added	Maxim Braverman		@RobertBryant I was not accurate. What one needs is that the matrices with respect to some coordinate system commute. Then setting $A(t)= (1-t) A_0+tA_1$ and $G(t)=e^{A(t)}$ we get $\partial G/\partial x^j= e^{A}\partial A/\partial x^j$ and the Christoffel symbols are linear in $A(t)$.
Nov 24, 2016 at 13:24	comment	added	Robert Bryant		@MaximBraverman: Your argument cannot be right because any two metrics can be simultaneously diagonalized in some frame, and hence their matrices will commute.
Nov 24, 2016 at 13:08	comment	added	Maxim Braverman		Your argument can be generalized. Suppose in some frame the matrices which represent $g_1$ and $g_0$ commute. Let $G$ and $G_0$ be these matrices. Then we can write $G_1= e^{A_1}$, $G_0= e^{A_0}$ where $A_1$ and $A_0$ a symmetric matrices which commute which each other. Then basically repeating your argument we obtain that $\nabla_t$ is the Levi-Civita connection of $G_t:= e^{(1-t)A_0+tA_1}$.
Nov 24, 2016 at 9:10	history	answered	Sebastian	CC BY-SA 3.0