Curvature sphere

Intuitively, curvature is the amount by which a geometric object such as a surface deviates from being a flat plane, or a curve from being straight as in the case of a line, but this is defined in different ways depending on the . A perfect sphere has constant curvature everywhere on the surface whereas the curvature on other surfaces is variable. For example on a rubgy ball the curvature is greatest at the ends and least in the middle. Measuring curvature at a point using curves through that point on the surface will not work. Introduction: The sphere has the interesting property that is Gaussian and mean curvatures are constant.

Because of its remarkable symmetry, though, this is hardly surprising. Note that, despite its relatively simple appearance, its parameterization in rectangular coordinates is somewhat complicated. Incidentally, Helgason defines the curvature of a 2-dimensional manifold by.

For the 2- sphere of radius R we have A . Unit speed spherical curve curvature. Geodesic curvature of sphere parallels. Curvature of a curve lying on a sphere. Weitere Ergebnisse von math. The Gaussian curvature can be given entirely in terms of the first fundamental form.

Alternatively, we could ask for 2-dim spaces of constant curva- ture. Computing the metric for a general 2-geometry, then imposing constant curvature gives a set of differential equations that will lead to this form. One definition of curvature starts by carrying a vector by parallel transport. Some new differentiable sphere theorems are obtained via the Ricci flow and stable currents. Parts of a plane are flat, Gauss and mean curvature are zero.

Cylinders and cones have Gauss curvature zero (since one principal curvature is zero). Spheres of radius r have mean . The curvature sphere provides information of the turning rate at a point on the curve. The center of the curvature sphere is always on the positive direction of the normal vector. As a result, the larger the curvature sphere , the smaller . Examples of constant mean curvature immersions of the 3- sphere into euclidean 4-space. A sphere theorem for three dimensional manifolds with integral pinched curvature.

Vincent Bour and Gilles Carron. It is obtained as the quotient of a certain isometric action of Sp(1) on Sp(2), and hence as a riemannian submersion from Sp(2). Mean curvature is one of the simplest and most basic of local differential geometric invariants. Therefore, closed hypersurfaces of constant mean curvature in euclidean spaces of high dimension are basic objects of fundamental importance in global differential geometry.

Before the examples of this paper, the only . Ezin, Scalar curvature functions in a conformal class of metrics and conformal transformations, in preparation. J Escobar, R SchoënConformal metrics with prescribed scalar curvature. Y Chang, P YangConformal deformations of metrics on S 2.