Normal fields of geodesic spheres

This question is related to this one ( I’ve asked at math.stackexchange. Let (M,g) be a compact Riemannian manifold with no conjugate points and (˜M,˜g) its universal covering. Let ˆg the Sasaki metric on TM−{0} and dˆg its associated distance function. Fix ˜p∈˜M and R=1. Let ˜H:=˜M−¯B1(p). For x∈˜H, consider the geodesic sphere centered at … Read more

Are ultralimits the Gromov-Hausdorff limits of a subsequence?

Let (Mi,pi) be a sequence of n-dimensional Riemannian manifolds with lower Ricci curvature bound −1. Fix a non-orincipal ultrafilter and let X be the ultralimit of the sequence. Does there exists a p∈X and subsequence of (Mi,pi) converging to (X,p) in the pointed Gromov-Hausdorff sense? Answer AttributionSource : Link , Question Author : dg.jan , … Read more

Osculating ellipsoids

Let K be a given smooth, origin-symmetric, strictly convex body in n dimensional euclidean space. At every point x on the boundary of K there exists an origin-symmetric ellipsoid Ex that touches x of second-order, the osculating ellipsoid at x. Denote the family of osculating ellipsoids by F:={Ex:x∈boundary of K}. Moreover, set G:={TE:E∈F & T∈SL(n)}. Is it true … Read more

Hodge-Weil Formula for Quaternionic-Kähler manifold

Let M be a quaternionic-Kähler manifold, with fundamental form ω, and let L be the Lefschetz operator of ω. In the Kähler and, more generally, symplectic cases, there is a mysterious relation between the Lefschetz sl2-decomposition and the Hodge operator due to Weil (see Huybrechts 1.2.31) Does there exist a quaternionic-Kähler analogue? Answer AttributionSource : … Read more

Faster (than normal) convergence of the normalized Ricci flow on surfaces

Consider a compact surface M of genus γ>1 (I am using the more usual letter “g” to denote metric), and the normalized Ricci flow on it. It is known that at time t, the scalar curvature R satisfies |R−r|<Cert, where r=∫MRdμ∫Mdμ is the average scalar curvature of M, and C is a constant depending only … Read more

Laplacian Spectra on Nearly Nodal Riemann Surfaces

Consider a family of complex curves C→D such that the central fibre is a nodal Riemann surface while other fibres are smooth Riemann surfaces. We choose a family of conformal metrics by restricting a smooth metric on C. So near the nodes (with local models xy=t, where t is the coordinate on D), the metric … Read more

smooth topos as generalized smooth space

I’m interested in generalized smooth spaces. I know there are several spaces such as Deffeological space, Frölicher space, Chen space, etc… and there are some papers compare them. However, I haven’t found one which compares smooth topos to them. (I don’t have literacy to do it on my own yet.) What are characteristics of smooth … Read more

Is positively curved Alexandrov surface isometrically embeddable in R3\mathbb R^3?

I guess it is not. The example I have in mind is: X2 is the spherical suspension of a circle S1(t) of length 0<t<2π. Then X has constant curvature =1 except at two suspension points, say N and S. But I cannot convince myself, since it seems this manifold can be approximated by a sequence … Read more

Obstructions to symplectically embedding compact manifolds of dimension 44 or higher

It is known in Li’s paper ( that in compact symplectic manifolds (X2n,ω) of dimension at least 2n≥4, an immersed symplectic surface represents a 2-homology class as long as that homology class has positive symplectic area. When 2n≥6, this immersion may even be taken to be an embedding. My question regards finding symplectically embedded submanifolds … Read more