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

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

Geodesic balls in warped product spaces

Let $g_S$ be a Riemannian metric on the $n$-dimensional sphere $S^{n}$ and consider the space $M=(0,a)\times S^{n}$ with the warped metric $g=dt^2+f(t)^2g_S$, where $f\colon [0,a)\to \mathbb{R}$ is a smooth function such that $$f(0)=0\,,\quad f(t) >0\quad t\in (0,a)\,.$$ I would like to understand which are the assumptions on $f$ in order to add a … 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

classification of homogenous complex manifolds

Suppose X is a complex manifold (doesn’t assume it’s Kahler), and it’s holomorhpic automorphism group is transitive. My question is that is there any classification of those manifolds ? Answer AttributionSource : Link , Question Author : user42804 , Answer Author : Community

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

Harmonicity on semisimple groups

I asked this on Math.SE and got no answer, so I’ll try my luck here. Let G be a semisimple real Lie group, U(g) its universal enveloping algebra, let Ω be the Casimir element in U(g) and let f be a smooth (or analytic) real-valued function on G. We then have the following notions 1) … Read more

Property of flat affine connection

Let M be a smooth simply-connected manifold. Let ∇ be a flat, symmetric connection on M. Let p∈M and let v,w∈TpM belong to a normal neighborhood, such that the ∇-geodesic triangle with vertices p, exppv and expp(v+w) is in M. I am looking for a proof that expp(v+w)=expexppv(Πexppvpw), where Πqp denotes the (path-independent) parallel transport … Read more

A quantity associated with a Riemannian surface

Assume that $E$ is a Riemannian vector bundle, then its structure group is reduced to $O(n)$. Then the structure group of $E \oplus E$ is reduced to $D(O(n) \oplus O(n)) \subset Sp(2n)$ where $$D(O(n) \oplus O(n))=\{ A\oplus A\mid A \in O(n)\}$$ So there is a natural symplectic structure $\omega$ on each fiber of \$E … Read more

Integral of second fundamental form

Let us have Riemannian manifold M with boundary N. Let F be an immersion, such that F:N→M and B be a second fundamental form on N relative to F. And let f be a function on N. How we can calculate integral ∫NB(grad(f),grad(f))Ω, where Ω is a volume form on N? Is there exist any … Read more