‎let $m^n$ be an $n(ngeq 3)$-dimensional complete connected and‎ ‎oriented spacelike hypersurface in a de sitter space or an anti-de‎ ‎sitter space‎, ‎$s$ and $k$ be the squared norm of the second‎ ‎fundamental form and gauss-kronecker curvature of $m^n$‎. ‎if $s$ or‎ ‎$k$ is constant‎, ‎nonzero and $m^n$ has two distinct principal‎ ‎curvatures one of which is simple‎, ‎we obtain some‎ ‎charact...

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.

‎Let $M^n$ be an $n(ngeq 3)$-dimensional complete connected and‎ ‎oriented spacelike hypersurface in a de Sitter space or an anti-de‎ ‎Sitter space‎, ‎$S$ and $K$ be the squared norm of the second‎ ‎fundamental form and Gauss-Kronecker curvature of $M^n$‎. ‎If $S$ or‎ ‎$K$ is constant‎, ‎nonzero and $M^n$ has two distinct principal‎ ‎curvatures one of which is simple‎, ‎we obtain some‎ ‎charact...

Second-order optimization methods such as natural gradient descent have the potential to speed up training of neural networks by correcting for the curvature of the loss function. Unfortunately, the exact natural gradient is impractical to compute for large models, and most approximations either require an expensive iterative procedure or make crude approximations to the curvature. We present K...

Let K be a convex body in R. A random polytope is the convex hull [x1, ..., xn] of finitely many points chosen at random in K. E(K,n) is the expectation of the volume of a random polytope of n randomly chosen points. I. Bárány showed that we have for convex bodies with C3 boundary and everywhere positive curvature c(d) lim n→∞ vold(K)− E(K,n) ( vold(K) n ) 2 d+1 = ∫ ∂K κ(x) 1 d+1 dμ(x) where κ(...