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

Classification of manifolds with Ric≥0{\rm Ric}\geq 0 wrt fundamental group

Note that n-manifolds M with Ric≥0 has a fundamental group of polynomial growth of degree ≤n (proof : use Bishop volume theorem). (Here a group Γ is said to have polynomial growth of degree ≤n if for any system of generators S there is an a>0 s.t. ϕS(s)≤asn where ϕS(s) is the number of elements … Read more

Compressing a hypersurface on the sphere

Let Mn be a compact, connected, orientable hypersurface of the unit sphere Sn+1⊂Rn+2. Suppose M is contained in the northern hemisphere Sn+1+ and has nonzero principal curvatures everywhere, i.e., has nonzero gaussian curvature everywhere. It seems to me that if we compress M somehow in the direction of the north pole, its principal curvatures will, … Read more

Limit cycles of quadratic systems and closed geodesics(Finitness of H(2)H(2))

This question is inspired by this answer to the question Finding a 1-form adapted to a smooth flow. Assume that V is a polynomial vector field of degree 2 as follows:{x′=P(x,y)y′=Q(x,y)(V) where P(x,y),Q(x.y) are polynomials of degree 2 in x,y with P(0,0)=Q(0,0)=0. We define the planar algebraic curve C={(x,y)∣yP−xQ=0} The above linked answer shows that: … Read more

Positive and non-negative sectional curvature of semi-riemannian metrics

I am currently studying the techniques related to geometry of positive and non-negative sectional curvature of Riemannian metrics. In particular, I have done some work with Cheeger deformations. I started to wonder: On the context of semi-Riemannian metrics, is there restrictive conditions for positive and non-negative sectional curvature? (I know that there are restrictions on … Read more

Flatness equivalence

Let π:E→M be a complex vector bundle and H a hermitian metric over it. If D is a connection over E, using the metric H, we can decompose it as: D=DH+ϕ Where DH is an unitary connection and ϕ is Hermitian 1-form with values in End(E). This can be done locally by taking the 1-form … Read more

Area lower bound given a mean curvature upper bound?

If Σ is a smooth embedded closed hypersurface in Rn with (normalized) mean curvature H≤1 (the mean curvature of the unit sphere), then its ((n−1)-dimensional) area is at least the area of the unit sphere in Rn. This is proved in Chavel’s book “Isoperimetric inequalities” Theorem II.1.3. The proof uses the idea that by restricting … Read more

Non-commutative analogue of a certain fact in differential geometry

In the literature, is there a non-commutative analogue of the fact that every Riemannian manifold whose isometry group has sharp dimension must be a constant curvature manifold? Answer AttributionSource : Link , Question Author : Ali Taghavi , Answer Author : Community

Parallel transport of vector along piecewise smooth loop on high-dimensional manifold

In this question, it was discussed that the rotation of a vector that is parallely transported along a piecewise smooth simple loop γ in a 2-dimensional manifold is exactly the integral of the gaussian curvature over the interior of the loop. The proof seems to be similar to the one of the 2-dimensional Gauss-Bonnet … Read more

Curvature universal abelian variety

I am reading N.Mok’s paper “Aspects of Kähler Geometry on Arithmetic varieties”, I am especially interested in the computation of the curvature for the space Hg×Cg, where Hg denotes Siegel’s upper half space of g×g symmetric complex matrices of positive definite imaginary part. One can put a 1-parameter family of Kähler metrics on Hg×Cg defied … Read more