What is a Riemannian metric? I have just started reading 'Riemannian Geometry' using primarily do Carmo's text and I've got the idea that to introduce a notion of distance to be able to talk about length of curves on smooth manifolds, angles between curves we introduce something called a 'Riemannian metric'.
It's definition on the other hand, seems very unintuitive(Carmo's definition doesn't involve anything tensor related). I guess it involves an inner product in it because we want to measure length of tangent vectors (but why?) and what better way to measure it than using inner product…
To be able to proceed further into the theory and make sure I can make sense of the things, I think it's important I understand the definition very well but after reading it again and again, I'm not sure if I 'get' it. I would love it if someone could give me an explanation for it.
Also, if we want to give a metric structure (to be able to talk about all things distance related) then why not define a metric (the topological one, satisfying positivity, symmetry, triangle inequality) rather than this?
Or maybe I'm confused and Riemannian metric as we've defined is actually that metric(topological one) and together with it, our smooth manifolds becomes a metric space? If that's the case, wouldn't the theory of Riemannian Geometry become somewhat easy as we already know a lot about metric spaces?(I seriously think, that that is not the case)
Thanks a lot for reading and answering/commenting (on) my post!
Best Answer
The idea is to equip the tangent space $T_p M$ at $p$ to the manifold $M$ with an inner product $\langle -, - \rangle$ , in such a way that these inner products vary smoothly as $p$ varies on $M$.
It is then possible to define the length of a curve segment on a $M$ and to define the distance between two points on $M$.
In terms of the definition of a Riemannian metric, given a smooth $n$-dimensional manifold, $M$, a Riemannian metric on $M$ (or $TM$) is a family, $(\langle -, - \rangle_p )_{p \,\in M}$, of inner products on each tangent space, $T_p M$, such that $\langle -, - \rangle$ depends smoothly on $p$.