Apr 3, 2016 · Indeed the metric h h is intrinsically defined by the restriction of g g to the submanifold L L, and there are infinitely many ways to describe a fixed submanifold L L as the level set of a regular function.

Oct 12, 2016 · Good afternoon, There is an example of a Riemannian metric on S2 ×S2 S 2 × S 2 of nonnegative sectional curvature that is not a product metric. I know there is one; however, I cannot find a specific reference. Any suggestions?

Let M be a compact subset of R2 with smooth boundary, and let g be a Riemannian metric on M. If g ′ is another Riemannian metric which is "close" to g, then they should have almost identical curvature profiles. I would like to prove a concrete estimate on the total difference of their curvatures in terms of the distance of g ′ to g.

Sep 25, 2017 · The Riemannian metric induced on TSn T S n from the Euclidean metric on Rn+1 ×Rn+1 R n + 1 × R n + 1 is Euclidean on each fiber TxSn T x S n (fix x x), and the metric on the zero section is the usual one (set ξ = 0 ξ = 0).

This is the simplest case of a question that's been bugging me for a while: say we have a Riemannian metric in polar coordinates on a $(2-d)$ surface: $$ g=dr^2+f^2(r, \\theta;)d\\theta^2, $$ such th...

I'm an old-fashioned differential geometer, so I don't understand all of this. Any chance you could explain how you define a Riemannian or Kahler metric as "something global and algebraic, free from messy local expressions"?

Is there a good "intrinsic" way of defining a metric on G(k, n) G (k, n) or more generally, any flag variety of (possibly partial) flags in Cn C n, that generalizes the Fubini–Study metric? Edit: I have streamlined the definition of the metric from the original version of the question.

Apr 14, 2023 · Suppose I have a smooth (high dimensional) Riemannian manifold, and fix a base point x0 x 0. Can I pick a coordinate system near x0 x 0 such that the metric is real analytic in some of the variables (maybe just in the first variable)? I know that I cannot necessarily make it real analytic in all the variables, but is it possible to always make it partially real analytic?

The metric of a Riemannian manifold determines the shortest distance between any two points. I assume the reverse holds? That is, if you are given the shortest distance d (x,y) between every pair of points of a manifold M, the metric for M is determined?

This has been studied by various authors, and e.g. Nikolaev proved that if ρ ρ is a length metric on a closed topological manifold X X of curvature bounded above and below in comparison sense, then X X has a C1,α C 1, α Riemannian metric that induces ρ ρ.