Normal fields of geodesic spheres

This question is related to this one (https://math.stackexchange.com/questions/1383511/normal-curvature-of-geodesic-spheres) 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 (http://arxiv.org/pdf/0812.4929v1.pdf) 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

Action of orthogonal group on the free Lie algebra

This question is somewhat related and inspired by this post of professor Montgomery. The free Lie algebra L(V) generated by an r-dimensional vector space V is, in the language of https://en.wikipedia.org/wiki/Free_Lie_algebra, the free Lie algebra generated by any choice of basis e1,…,er for the vector space V. (Work over the field R or C, whichever … Read more