Advanced Search

We will prove that if an open subset of $ℂP^n$ is isometrically immersed into $ℂP^m$, with $m<(4/3)n-2/3$, then the image is totally geodesic. We will also prove that if an open subset of $ℍP^n$ isometrically immersed into $ℍP^m$, with $m<(4/3)n-5/6$, then the image is totally geodesic.

As a numerical method for solving ordinary differential equations $y^{\text{'}}=f(x,y)$, the improved Euler method is not assumed to give exact solutions. In this paper we classify all cases where this method gives the exact solution for all initial conditions. We reduce an infinite system of partial differential equations for $f(x,y)$ to a finite system that is sufficient and necessary for the improved Euler method to give the exact solution. The improved Euler method is the simplest explicit second order Runge-Kutta method....