Background: Scoliosis is a health problem that causes a side-to-side curvature in the spine. The curvature may have an “S” or “C” shape. To evaluate scoliosis, the Cobb angle has been commonly used. However, digital image processing allows the Cobb angle to be obtained easily and quickly, several researchers have determined that Cobb angle contains high variations (errors) in the measurements. ...

The Jacobi metric introduced in mathematical physics can be used to analyze Hamiltonian Monte Carlo (HMC). In a geometrical setting, each step of HMC corresponds to a geodesic on a Riemannian manifold with a Jacobi metric. Our calculation of the sectional curvature of this HMC manifold allows us to see that it is positive in cases such as sampling from a high dimensional multivariate Gaussian. ...

We provide new examples of manifolds which admit a Riemannian metric with sectional curvature nonnegative, and strictly positive at one point. Our examples include the unit tangent bundles of CP, HP and CaP, and a family of lens space bundles over CP.

In this note we study isometric immersions of Riemannian manifolds with positive Ricci curvature into an Euclidean space.

We prove that the connected sums CP2#CP2 and CP2#CP2#CP2 admit self-dual metrics with positive Ricci curvature. Moreover, every self-dual metric of positive scalar curvature on CP2#CP2 is conformal to a metric with positive Ricci curvature. ∗Supported in part by NSF grant DMS-9204093

In this paper we address the issue of positive scalar curvature on oriented nonspin compact manifolds whose universal cover is also nonspin. We provide a conjecture for an obstruction to such curvature in this venue that takes into account all the data known to date. The conjecture is proved for a wide class of closed manifolds based on their fundamental group structure.

We show that the unit tangent bundle of S4 and a real cohomology CP 3 admit Riemannian metrics with positive sectional curvature almost everywhere. These are the only examples so far with positive curvature almost everywhere that are not also known to admit positive curvature. AMS Classi cation numbers Primary: 53C20 Secondary: 53C20, 58B20, 58G30

A closed Riemannian n-manifold M with Ricci M > n 1 has diameter < 7r ([M]), and equality holds only if M is isometric to the unit sphere S ~ ([Cn]). Given these results, it is natural to ask whether M is diffeomorphic to S n if the diameter of M is almost 7r. This question was answered negatively by Anderson and Otsu, who showed that even the topology of such a space can be different from the ...

Examples of almost-positively and quasi-positively curved spaces of the form M = H\((G, h) × F ) were discovered recently [9],[8]. Here h is a left-invariant metric on a compact Lie group G, F is a compact Riemannian manifold on which the subgroup H ⊂ G acts isometrically on the left, and M is the orbit space of the diagonal left action of H on (G, h)×F with the induced Riemannian submersion me...

We prove that any simply connected and complete Riemannian manifold, on which a Randers metric of positive constant flag curvature exists, must be diffeomorphic to an odd-dimensional sphere, provided a certain 1-form vanishes on it. 1. Introduction. The geometry of Finsler manifolds of constant flag curvature is one of the fundamental subjects in Finsler geometry. Akbar-Zadeh [1] proved that, u...