What is the connection between a metric and a manifold? I am in process of reading a paper which contains something called a "Shahshahani Metric" which has uses in mathematical biology
https://www.google.ca/search?q=Shahshahani+metric&rlz=1C1CHWA_enCA601CA601&oq=Shahshahani+metric&aqs=chrome..69i57j0l5.3129j0j7&sourceid=chrome&es_sm=93&ie=UTF-8
But I failed to appreciate the link between this so called Shahshahani Metric and Shahshahani manifold
Can someone motivate the definition of a metric and how it relates to manifold by intuitively describe the connection between Riemannian metric and Riemannian manifold?
Can someone show me how a metric of a manifold can be used to provide some information?
 A: A riemannian metric on a manifold provides a way to measure lengths and angles between tangent vectors. This in turns gives ways to measure areas and volumes, as well as distance between points in a manifold.
A riemannian manifold is a manifold supporting a given riemannian metric. Nice manifolds, satisfying some topological assumptions, admit such a geometric structure.
Any good book on manifolds will describe riemannian metrics at some point. One book that I really like is Lee's introduction to smooth manifolds.
A: A smooth manifold is a topological construct on which the only structure is the open sets and the smooth atlas. This is useful because this small amount of structure is enough to do a lot of sophisticated math with manifolds. The downside is that manifolds which seem like they ought to be different have indistinguishable topologies and smooth structures. There is no difference between, say, a sphere, an ellipsoid, and the surface of a squashed up piece of clay. In order to determine a difference between each of these manifolds, we have to add more structure. That structure is the metric. Intuitively, it tells us how the way that two vectors on the manifold relate to each other depends on the position within the manifold--it defines an inner product between vectors. This allows us to determine things like distance, angle, shortest paths, curvature, and shape whereas the smooth manifold structure told us none of this. The metric is often called the Riemannian metric, and a Riemannian manifold is a smooth manifold with a Riemannian metric associated with it.
