Consider a two-dimensional sphere with a Riemannian metric of total area $4\pi$. Does there exist a subset whose area equals $2\pi$ and whose boundary has length no greater than $2 …

Given a positive real number $l$. Does there exist a closed hyperbolic surface $X$ so that injectivity radius not less than $l$?

I'm currently learning some stuffs about systolic inequalities. While reading the relevant sections (p329 to 340) in Berger's Panoramic View of Riemannian Geometry, I noticed a gap …

If the Gaussian curvature of the metric $g= f^2(x,y)(dx^2+dy^2)$ is nonzero then $f$ cannot be constant. This can be expressed by stating that the (probabilistic) variance $Var(f) …

I have been reading an article, and I am not clear on what he does in this one part. He starts with a Riemmannian manifold $\Sigma$ that is $\mathbb{H}^2/\Gamma$, a quotient of th …

it is well known that if the fundamental group of a manifold M is residualy finite then for every point p in M and every epsilon positive there is a finite covering such that for …

It's all in the title basically. There's an interesting topic called systolic geometry that has grown a lot in the past 30 years, with a (first?) textbook on the subject by M.Katz …