I've done coursework in differential geometry and stochastic calculus, but I haven't formally seen any connections between the two. I have read that both information geometry and stochastic differential geometry are two different approaches to link the two fields; however, from searching around the internet I haven't settled on any good learning resources that I find accessible.

My questions

Any recommendations on a readable book/paper recommendations that specifically link stochastic calculus to manifolds? I'm aware of Emery's book. If there was something that was perhaps slightly more readable, albeit more informal, that would be ideal.

Is there a formulation of Feynman-Kac on manifolds?

Is there a formulation of the HJB PDE for stochastic control problems on manifolds?