Suppose we have a Riemannian manifold $M$ with two complete metrics $g_0$, $g_1$ such that their curvatures are bounded and injectivity radii are bounded away from zero.
My goal is to find a uniform lower bound away from zero for the injectivity radius of their convex sum $$g_t:=(1-t)g_0+tg_1$$ for all $t\in[0,1]$.
What I want to try is that given $p\in M$ and $v\in T_pM$, consider $exp_p^{g_t}(v)=exp_p^{(1-t)g_0+tg_1}(v)$.
I wonder if there are properties of exponential map which will be useful. For example, $$exp_p^{tg}(v)=exp_p^{g}(tv)?$$ or $$exp_p^{g_1+g_2}(v)=exp^{g_2}_{exp^{g_1}_p(v)}(Dexp_p^{g_1}(v))?$$
Also, It would be very helpful if I could get a reference for this topic.