Skip to main content
1 vote
Accepted

Some errata in Petersen's Riemannian Geometry book 3rd?

The Hessian of the distance function $r$ can never be completely positive definite. Indeed, since $|\nabla r|^2 = 1$ is constant, differentiating in the direction of $\nabla r$ yields $$ \mathrm{Hess}\...
Didier's user avatar
  • 19.7k
1 vote
Accepted

On writing integration on Riemannian submanifolds in terms of exponential map

The answer is no, and the reason appears if you write properly what it means that "$N$ is determined by the exponential map". In rigorous terms, you are asking the exponential map $\exp_x$ ...
Didier's user avatar
  • 19.7k
1 vote
Accepted

Cheeger-Gromov limit of scalings of a noncompact Riemannian manifold

Take the quotient $M$ of the hyperbolic (upper half) plane by the cyclic group generated by the translation $z\mapsto z+1$. Take points $p_k\in M$ which are projections of $k\sqrt{-1}$. Use $r_k=\sqrt{...
Moishe Kohan's user avatar
1 vote

Generator of translations in a Riemannian manifold

The Taylor formula has nothing to do with metrics. The partial derivatives act as generators along a a mesh considered as a flat, cartesian lattice. Any function like e.g. a metric translates by the ...
Roland F's user avatar
  • 3,259

Only top scored, non community-wiki answers of a minimum length are eligible