This paper presents an algorithm for estimation of local curvature from gradients of a tensor field that represents local orientation. The algorithm is based on an operator representation of the orientation tensor, which means that change of local orientation corresponds to a rotation of the eigenvectors of the tensor. The resulting curvature descriptor is a vector that points in the direction ...

Recent work in even-dimensional conformal geometry [1],[3], [4], [6] has revealed the importance of conformally invariant powers of the Laplacian on a conformal manifold; that is, of operators Pk whose principal part is the same as ∆ k with respect to a representative of the conformal structure. These invariant powers of the Laplacian were first defined in [5] in terms of the Fefferman-Graham [...

Suppose that Σ = ∂M is the n-dimensional boundary of a connected compact Riemannian spin manifold (M, 〈 , 〉) with nonnegative scalar curvature, and that the (inward) mean curvature H of Σ is positive. We show that the first eigenvalue of the Dirac operator of the boundary corresponding to the conformal metric 〈 , 〉H = H 〈 , 〉 is at least n/2 and equality holds if and only if there exists a non-...

We study constant mean curvature 1/2 surfaces in H × R that admit a compactification of the mean curvature operator. We show that a particular family of complete entire graphs over H admits a structure of infinite dimensional manifold with local control on the behaviors at infinity. These graphs also appear to have a half-space property and we deduce a uniqueness result at infinity. Deforming n...

Topological energy bands have important geometrical properties described by the Berry curvature. We show that the Berry curvature changes the hydrodynamic equations of motion for a trapped Bose-Einstein condensate, and causes significant modifications to the collective mode frequencies. We illustrate our results for the case of two-dimensional Rashba spin-orbit coupling in a Zeeman field. Using...

A method of Hurwitz-Radon Matrices (MHR) is proposed to be used in parametrization and interpolation of contours in the plane. Suitable parametrization leads to curvature calculations. Points with local maximum curvature are treated as feature points in object recognition and image analysis. The matrices are skew-symmetric and possess columns composed of orthogonal vectors. The operator of Hurw...

Using Fefferman’s classical result on the boundary singularity of the Bergman kernel, we give an analogous description of the boundary behaviour of various related quantities like the Bergman invariant, the coefficients of the Bergman metric, of the associated Laplace-Beltrami operator, of its curvature tensor, Ricci curvature and scalar curvature. The main point is that even though one would e...

The phase structure of Nambu-Jona-Lasinio model with N-component fermions in curved space-time is studied in the leading order of the 1/N expansion. The effective potential for composite operator ¯ ψψ is calculated by using the normal coordinate expansion in the Schwinger proper-time method. The existence of the first-order phase transition caused by the change of the space-time curvature is co...

Let s ≥ 2. We construct Ricci flat pseudo-Riemannian manifolds of signature (2s, s) which are not locally homogeneous but whose curvature tensors never the less exhibit a number of important symmetry properties. They are curvature homogeneous; their curvature tensor is modeled on that of a local symmetric space. They are spacelike Jordan Osserman with a Jacobi operator which is nilpotent of ord...

An algebraic curvature tensor is called Osserman if the eigenvalues of the associated Jacobi operator are constant on the unit sphere. A Riemannian manifold is called conformally Osserman if its Weyl conformal curvature tensor at every point is Osserman. We prove that a conformally Osserman manifold of dimension n 6= 3, 4, 16 is locally conformally equivalent either to a Euclidean space or to a...