The space of direction at a point in an infinite dim Alexandrov space can be compact?Please give examples or prove it's wrong.

As we know,if M is an Alexandrov space with sec>=1,then the cone over M has sec>=0.What if when M is a cuboid with side length r1,...,rn,dia(M)<=π,then the cuvature of the cone …

M is a warped product $M = R \times {}_{{e^{2t}}}N$.N is the spherical suspension over$R{P^2}$.N is an Alexandrov space but not a manifold.So is M.Is there a harmonic function on M …

Let $M$ be a metric space, $A$ a subset of $M$. The reach (defined by Federer) of $A$ in $M$ is the largest $r_0\ge 0$ such that if $x\in M$ and the $d(x, A)< r_0$, then $A$ con …

Let M be a complete Riemannian manifold,suppose there is a harmonic function f on M.It was first observed by Yau that$${\left| {{\nabla ^2}f} \right|^2} \ge \frac{n}{{n - 1}}{\left …

M is a hyperbolic cone over an n-1 dim Riemannian manifold N with $Ric(N) \ge - \left( {n - 2} \right)$ ie $M = R \times {}_{\cosh \left( t \right)}N$,Surely N is an Alexandrov sp …

M is a Riemannian manifold with $$Ric \ge - (n - 1)$$,f is a harmonic function on M,let $$h = |\nabla f|$$.By choosing an orthonormal basis such that $$|\nabla f|{e_1} = \nabla f, …

could someone give some examples :nonRiemannian manifold Alexandrov space with sec>=-1 and the first eigenvalue equal to (n-1)^2/4

I want to prove some Alexandrov space M is parabolic cone X x R.Since Alex has no Riemannian metric,so how to do?Is there any (triangle) formula about the relation of distance of t …

To need not worry about the possibly broadest context let: $X$ be an Alexandrov's space with lower curvature bound and $C$ be a totally convex subset, i.e. for any $x,y \in C$ and …

X is an n-dim positively curved manifold and Y is a totally geodesic submanifold of codimension 1.Then cutting along Y we get n-dim positively curved manifolds without boundary,by …

I heard a conjecture "3-dim positively curved Alexandrov space is of the form S^3/J.(I cannot make sure my statement is accurate). What is the classification of n-dim positively cu …

In the conference "on Manifolds with Non-negative Sectional Curvature" held in 2007, Problem 6 is: Extend the Wilking Connectivity Theorem to Alexandrov spaces, i.e. if $X$ is …

Let $X$ be an $n$-dimensional Alexandrov space, $f: X\to \mathbb R$ a semi-concave function. One can define (see for example Petrunin "Semi-concave functions in Alexandrov geometry …

Could you give examples of totally geodesic subset of codim>1 in positively curved Alexandrov space?