We show that the " Taub-bolt instantons " are the only Riemannian, Ricci-flat and asymptotically locally flat (ALF) C 2-metrics on a 4-dimensional, complete, simply connected man-ifold having (at least) a one-parameter group of isometries which are compatible with ALF, have trajectories of bounded length at infinity and no isolated fixed points. Attempts of estimating the path integral of Quant...

The theory of monotone Riemannian metrics on the state space of a quantum system was established by Dénes Petz in 1996. In a recent paper he argued that the scalar curvature of a statistically relevant – monotone – metric can be interpreted as an average statistical uncertainty. The present paper contributes to this subject. It is reasonable to expect that states which are more mixed are less d...

A class of elliptic-hyperbolic equations is placed in the context of a geometric variational theory, in which the change of type is viewed as a change in the character of an underlying metric. A fundamental example of a metric which changes in this way is the extended projective disc, which is Riemannian at ordinary points, Lorentzian at ideal points, and singular on the absolute. Harmonic fiel...

In this paper we study critial isometric and minimal isometric embeddings of classes of Riemannian metrics which we call quasi-κ-curved metrics. Quasi-κ-curved metrics generalize the metrics of space forms. We construct explicit examples and prove results about existence and rigidity. Introduction Definition: Let (M, g̃) be a Riemannian manifold. We will say g̃ is a quasi-κcurved metric if there ...

In computational anatomy, organ’s shapes are often modeled as deformations of a reference shape, i.e., as elements of a Lie group. To analyze the variability of the human anatomy in this framework, we need to perform statistics on Lie groups. A Lie group is a manifold with a consistent group structure. Statistics on Riemannian manifolds have been well studied, but to use the statistical Riemann...

We present evolution equations for a family of paths that results from anisotropically weighting curve energies in non-linear statistics of manifold valued data. This situation arises when performing inference on data that have non-trivial covariance and are anisotropic distributed. The family can be interpreted as most probable paths for a driving semi-martingale that through stochastic develo...

We consider the space of Kähler metrics as a Riemannian submanifold of the space of Riemannian metrics, and study the associated submanifold geometry. In particular, we show that the intrinsic and extrinsic distance functions are equivalent. We also determine the metric completion of the space of Kähler metrics, making contact with recent generalizations of the Calabi–Yau Theorem due to Dinew a...

A Riemannian metric on a manifold M induces a family of Riemannian metrics on the loop space LM depending on a Sobolev space parameter. In Part I, we compute the Levi-Civita connection for these metrics. The connection and curvature forms take values in pseudodifferential operators (ΨDOs), and we compute the top symbols of these forms. In Part II, we develop a theory of Chern-Simons classes CS ...

Conformal geometry is in the core of pure mathematics. It is more flexible than Riemaniann metric but more rigid than topology. Conformal geometric methods have played important roles in engineering fields. This work introduces a theoretically rigorous and practically efficient method for computing Riemannian metrics with prescribed Gaussian curvatures on discrete surfaces – discrete surface Ri...

We prove that all smooth sphere bundles that admit fiberwise 1/4-pinched metrics are induced bundles of vector bundles, so their structure groups reduce from DIFF(Sn) to O(n + 1). This result implies the existence of many smooth Sn-bundles over Sk that do not support strictly 1/4-pinched positively curved Riemannian metrics on their fibers.