I would like to find some references where there are specific computations and properties of rotationally symmetric riemannian manifolds, e.g. spectrum of the laplacian, schrödinger operators, examples of geometric flows, minimal manifolds in $\mathbb{R}^n$, stability, computations for spheres, catenoids, etc.
Edit: Some basic examples I know are from Grigor'yan's "Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds", section 3 and others examples treating radial solutions. It would be very useful for self study to find something more exhaustive or comprehensive in the direction of geometry for these families of examples.
Edit 2: As clarification, I am thinking on Riemannian manifolds $(M,g)$ such that in local polar/radial coordinates the metric has the form $g = dr^2 + a^2(r) d\theta^2$, cf. Grigor'yan, section 3, and surely others references (that I am precisely asking for). The other tensors and local formulas should normally depend only on the radial coordinate $r$ and the expression of $a$, and the study of many geometric PDEs simplify to ODEs. Thus some explicit computations become feasible and one can gain intuition studying the formulas obtained, e.g. to the case geodesic spheres (constant $r$), taking limits on $r$, etc.