Civita Connection Research Articles

Smooth geometric evolutions of hypersurfaces

We consider the gradient flow associated to the following functionals:¶¶\( {\cal F}_m(\varphi) = \int_M 1 + |\bigtriangledown^m\nu|^2\,d\mu \).¶The functionals are defined on hypersurfaces immersed in \( {\mathbb R}^{n+1} \) via a map \( \varphi : M \to {\mathbb R}^{n+1} \), where M is smooth closed and connected n-dimensional manifold without boundary. Here \( \mu \) and \( \bigtriangledown \) are respectively the canonical measure and the Levi—Civita connection of the Riemannian manifold (M,g), where the metric g is obtained by pulling back on M the usual metric of \( {\mathbb R}^{n+1} \) with the map \( \mu \). The symbol \( \bigtriangledown^m \) denotes the mth iterated covariant derivative and \( \nu \) is a unit normal local vector field to the hypersurface.¶Our main result is that if the order of derivation \( m \in {\mathbb N} \) is strictly larger than the integer part of n/2 then singularities in finite time cannot occur during the evolution.¶These geometric functionals are related to similar ones proposed by Ennio De Giorgi, who conjectured for them an analogous regularity result. In the final section we discuss the original conjecture of De Giorgi and some related problems.

On the Gauss–Bonnet Formula in Riemann–Finsler Geometry

Using Chern's method of transgression, the Euler class of a compact orientable Riemann–Finsler space is represented by polynomials in the connection and curvature matrices of a torsion‐free connection. When using the Chern connection (and hence the Christoffel–Levi–Civita connection in the Riemannian case), this result extends the Gauss–Bonnet formula of Bao and Chern to Finsler spaces whose indicatrices need not have constant volume.

Variational Restoration of Nonflat Image Features: Models and Algorithms

We develop both mathematical models and computational algorithms for variational denoising and restoration of nonflat image features. Nonflat image features are those that live on Riemannian manifolds, instead of on the Euclidean spaces. Familiar examples include the orientation feature (from optical flows or gradient flows) that lives on the unit circle S1 , the alignment feature (from fingerprint waves or certain texture images) that lives on the real projective line $\mathbb{RP}^1$, and the chromaticity feature (from color images) that lives on the unit sphere S2 . In this paper, we apply the variational method to denoise and restore general nonflat image features. Mathematical models for both continuous image domains and discrete domains (or graphs) are constructed. Riemannian objects such as metric, distance and Levi--Civita connection play important roles in the models. Computational algorithms are also developed for the resulting nonlinear equations. The mathematical framework can be applied to restoring general nonflat data outside the scope of image processing and computer vision.

HOLOMORPHIC AND KILLING VECTOR FIELDS ON COMPACT BALANCED HERMITIAN MANIFOLDS

We extend the classical theorems for nonexistence of Killing and holomorphic vector fields on compact Kähler manifolds to compact balanced Hermitian manifolds. We express the obstructions to the existence of Killing and holomorphic vector fields on compact balanced Hermitian manifolds in terms of the Ricci tensors of the Levi–Civita connection as well as in terms of the Ricci tensors of the Chern connection. We show that every affine vector field with respect to the Chern connection on a compact balanced Hermitian manifold is holomorphic.

When Do Some Geodesics of the Induced and the Metrical Connection Coincide?

AbstractIn affine differential geometry, three natural connections can be considered, namely the induced connection, the Levi Civita connection of the affine metric and the conjugate connection. Quadrics are characterized by the conditions that all the pre‐geodesies with respect to one of these connections are also pre‐geodesies with respect to the other two connections. In the present paper we describe 3‐dimensional affine hypersurfaces for which some of the geodesies or pre‐geodesies for the different connections coincide.

The Einstein action for algebras of matrix valued functions—Toy models

Two toy models are considered within the framework of noncommutative differential geometry. In the first one, the Einstein action of the Levi–Civita connection is computed for the algebra of matrix valued functions on a torus. It is shown that, assuming some constraints on the metric, this action splits into a classical-like, a quantum-like and a mixed term. In the second model, an analogue of the Palatini method of variation is applied to obtain critical points of the Einstein action functional for M4(R). It is pointed out that a solution to the Palatini variational problem is not necessarily a Levi–Civita connection. In this model, no additional assumptions regarding metrics are made.

On locally symmetric affine hypersurfaces

1. Introduction. In the present paper, we investigate 3-dimensional. locally strongly convex affine hyperspheres M in N, ~. We will also assume that the Levi Civita connection ~' of the affine metric h is locally symmetric. This problem is related to the following two questions proposed at the conference on affine differential geometry at Oberwolfach ([10]) in 1986:

Affine hypersurfaces with parallel cubic form

As is well known, there exists a canonical transversal vector field on a non-degenerate affine hypersurface M. This vector field is called the affine normal. The second fundamental form associated to this affine normal is called the affine metric. If M is locally strongly convex, then this affine metric is a Riemannian metric. And also, using the affine normal and the Gauss formula one can introduce an affine connection ∇ on M which is called the induced affine connection. Thus there are in general two different connections on M: one is the induced connection ∇ and the other is the Levi Civita connection of the affine metric h. The difference tensor K is defined by K(X, Y) = KXY — ∇XY — XY. The cubic form C is defined by C = ∇h and is related to the difference tensor by.

3-dimensional affine hypersurfaces in ℝ4 with parallel cubic form

In this paper, we study 3-dimensional locally strongly convex affine hypersurfaces in ℝ4. Since the publication of Blaschke’s book [B] in the early twenties, it is well-known that on a nondegenerate affine hyper-surface M there exists a canonical transversal vector field called the affine normal. The second fundamental form associated to the affine normal is called the affine metric. In the special case that M is locally strongly convex, this affine metric is a Riemannian metric. Also, using the affine normal, by the Gauss formula one can introduce an affine connection on M, called the induced connection ∇. So on M, we can consider two connections, namely the induced affine connection ∇ and the Levi Civita connection of the affine metric h.