In this paper, we prove the following two results: First, we study a class of conformally invariant operators P and their related conformally invariant curvatures Q on even-dimensional Riemannian manifolds. When the manifold is locally conformally flat(LCF) and compact without boundary, Q-curvature is naturally related to the integrand in the classical Gauss-Bonnet-Chern formula, i.e., the Pfaf...

In this paper we discuss the problem of isometric embedding of the surface of a rapidly rotating black hole in a flat space. It is well known that intrinsically defined Riemannian manifolds can be isometrically embedded in a flat space. According to the Cartan-Janet [1, 2] theorem, every analytic Riemannian manifold of dimension n can be locally real analytically isometrically embedded into E w...

A Gauss Equation is proved for subspaces of Alexandrov spaces of curvature bounded above by K. That is, a subspace of extrinsic curvature ≤ A, defined by a cubic inequality on the difference of arc and chord, has intrinsic curvature ≤ K +A. Sharp bounds on injectivity radii of subspaces, new even in the Riemannian case, are derived.

Let F : Σ n × [0, T) → R n+m be a family of compact immersed submanifolds moving by their mean curvature vectors. We show the Gauss maps γ : (Σ n , g t) → G(n, m) form a harmonic heat flow with respect to the time-dependent induced metric g t. This provides a more systematic approach to investigating higher codimension mean curvature flows. A direct consequence is any convex function on G(n, m)...

In this paper, we prove the following two results: First, we study a class of conformally invariant operators P and their related conformally invariant curvatures Q on even-dimensional Riemannian manifolds. When the manifold is locally conformally flat(LCF) and compact without boundary, Q-curvature is naturally related to the integrand in the classical Gauss-Bonnet-Chern formula, i.e., the Pfaf...

1. Introduction. Perhaps the most significant aspect of differential geometry is that which deals with the relationship between the curvature properties of a Riemannian manifold M and its topological structure. One of the beautiful results in this connection is the (generalized) Gauss-Bonnet theorem which relates the curvature of compact and oriented even-dimensional manifolds with an important...

In this paper, simple and explicit formulas for computing mean curvature vector and Gauss–Kronecker curvature for n-manifolds in Rn+m are derived. Using these formulas, we solve an open problem, proposed by Ron Goldman, about curvature formulas for implicit surfaces with higher co-dimensions. © 2006 Elsevier B.V. All rights reserved.

A new algorithm for mesh simplification with triangle constriction is presented in this paper. Constricting error defined by a combination of square volume error variation with constraint (SVEC), shape factor and normal constraint factor of triangle. Gauss curvature factor of each constricted triangle is used to distinguish strong feature triangle or nonstrong feature triangle. The triangle whi...

We give the best possible upper bound on the number of exceptional values and totally ramified value number of the hyperbolic Gauss map for pseudo-algebraic Bryant surfaces and some partial results on the Osserman problem for algebraic Bryant surfaces. Moreover, we study the value distribution of the hyperbolic Gauss map for complete constant mean curvature one faces in de Sitter three-space.

Corrections to solar system gravity are derived for f(G) gravity theories, in which a function of the Gauss-Bonnet curvature term is added to the gravitational action. Their effects on Newton’s law, as felt by the planets, and on the frequency shift of signals from the Cassini spacecraft, are both determined. Despite the fact that the Gauss-Bonnet term is quadratic in curvature, the resulting c...