Bi-invariant Riemannian metrics on So(n) - MathOverflow
2013年4月3日 · Defining the inner product - 1/2 tr(XY) on the lie algebra so(n)(skew symmetric matrices) is one way to introduce a bi-invariant metric on So(n) since the inner product is ad-invariant. Are there any
Dual Riemannian metric and the Dual Metric Form - MathOverflow
It was just a guess on my part, that no matter what nice structures you have in mind, the Kahlerian compatible triple (that is, the compatible metric, symplectic, and complex structures) and/or the Hamiltonian flows thereby associated to smooth functions, will be helpful in constructing them.
Riemannian metric of hyperbolic plane - MathOverflow
2015年9月10日 · Certainly the general idea goes back at least to Minkowski, and/or Poincare, maybe Beltrami. (Poincare's model of the hyperbolic plane had geodesics arcs of circles orthogonal to the boundary, while Beltrami's geodesics were …
Taylor expansion of the metric tensor in the normal coordinates
2020年12月23日 · I am looking for a reference with a Taylor expansion of the metric tensor in the normal coordinates. The coefficients should be written in terms of $\mathrm{Rm}, \nabla\mathrm{Rm}, \nabla^2\mathrm{Rm},\dots$ at the point.
Riemannian metric on a level set of a smooth function on a manifold
2016年4月3日 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Existence of incomplete Riemannian metrics - MathOverflow
2016年11月1日 · And how do I construct the Riemannian metric such that this curve has finite length? If the curve is a closed 1-dimensional submanifold, then I just pick a metric on it and extend it using a partition of unity argument, but what …
Changing coordinates so that one Riemannian metric matches …
In the excellent thread When is a Riemannian metric equivalent to the flat metric on $\mathbb R^n$? , Greg Kuperberg says: If remember correctly, there is a more general result due to somebody, that any two Riemannian manifolds are locally isometric if and only if their curvature tensors are locally the "same".
Make a Riemannian metric real analytic in some coordinates
2023年4月14日 · Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Riemannian metric induced by a metric - MathOverflow
Here is a closely related question that may have been what the OP was driving at. Suppose that you ONLY know a Riemannian manifold as a metric space---that is you know the point set and the distance between any two points, but you do not know …
How does curvature change under perturbations of a Riemannian …
$\begingroup$ Although I agree with the sentiment here (in the long run you really want to understand things and work as much as possible using a co-ordinate-free point of view), I think that a student should first understand how to derive the inequality above in the most direct way possible with minimal machinery.