Property of distance function with "smoothly" varying Riemannian metrics

Question

Let $(M,g)$ be a smooth and compact Riemannian manifold. Suppose I have a "smoothly varying" (precisely formulating this is part of the question) one parameter family of Riemannian metrics $(M,g_t)$ that depends on a parameter $t \in [0,1]$. What I want to know is whether the procedure that turns a Riemannian manifold into a metric space is suitably continuous in some sense. More precisely,

a) Can I prove that there are uniform bound upper and lower bounds on the diameter $D(M,g_t)$?

b) Does the distance function between two points $m_1,m_2$ vary continuously with t?

What I would ideally like is a book that discusses this. Though a clear answer either way would work too obviously.

***I'm a little worried I'm going to get yelled at for this question. I'm sure for a researcher in differential geometry this is a trivial question, but it isn't my area and I do not know where to look for this information. It seems to fall into a gray area between Math Stackexchange and here, so I'd be grateful for some indulgence.

Answer 1

About the formulation, I would see such a family as a smooth map from $[0,1]$ to the space of smoooth sections of $S^2T^*M$ which lands in the open set of sections with values in positively definite forms.

Next, about the diameter. Because $M$ is compact, you can find a Riemannian metric $g_M$ such that $g_t\le g_M$ for any $t\in [0,1]$ and, in particular, you have $\mathrm{diam}(M,g_t) \le \mathrm{diam}(M,g_M)$.

For question b, this is true as well. Given $x,y\in M$ and $t_0\in [0,1]$, consider a geodesic $\gamma_{t_0}$ between $x$ and $y$ relatively to $g_{t_0}$. The length of $\gamma_{t_0}$ with respect to $g_t$ varies smoothly with $t$, so you find that \begin{equation} (A) \quad \limsup_{t\to t_0} d_{g_t}(x,y) \le d_{g_{t_0}}(x,y) \end{equation}

Conversely, if $\gamma_t$ is geodesic of minimal length wrt $g_t$ between $x,y$, then one can choose a sequence $t_n \to t_0$ such that $\gamma_{t_n}'(0)$ converges to some unitary vector $X\in T_xM$. By the smooth dependence of the geodesic on the initial speed and the metric, the geodesics $\gamma_{t_n}$ will converge to the geodesic wrt $g_{t_0}$ with inital speed $X$. Therefore the length $d_{g_{t_n}}(x,y)=\ell(\gamma_{t_n}, g_{t_n})$ converges to $\ell(\gamma_{t_0}, g_{t_0}) \ge d_{g_{t_0}}(x,y)$, and ones gets the converse of the inequality (A) above.