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In affine three-space the Cayley surface is given by x_3=x_1x_2-1/3x_1^3 (1) (Nomizu and Sasaki 1994). The surface has been generalized by Eastwood and Ezhov (2000) to ...

The first type of tensor-like object derived from a Riemannian metric g which is used to study the geometry of the metric. Christoffel symbols of the first kind are variously ...

The geodesics in a complete Riemannian metric go on indefinitely, i.e., each geodesic is isometric to the real line. For example, Euclidean space is complete, but the open ...

The covariant derivative of a contravariant tensor A^a (also called the "semicolon derivative" since its symbol is a semicolon) is given by A^a_(;b) = ...

The directional derivative del _(u)f(x_0,y_0,z_0) is the rate at which the function f(x,y,z) changes at a point (x_0,y_0,z_0) in the direction u. It is a vector form of the ...

A dyadic, also known as a vector direct product, is a linear polynomial of dyads AB+CD+... consisting of nine components A_(ij) which transform as (A_(ij))^' = ...

G_(ab)=R_(ab)-1/2Rg_(ab), where R_(ab) is the Ricci curvature tensor, R is the scalar curvature, and g_(ab) is the metric tensor. (Wald 1984, pp. 40-41). It satisfies ...

An elliptic fixed point of a differential equation is a fixed point for which the stability matrix has purely imaginary eigenvalues lambda_+/-=+/-iomega (for omega>0). An ...

Two metrics g_1 and g_2 defined on a space X are called equivalent if they induce the same metric topology on X. This is the case iff, for every point x_0 of X, every ball ...

Euclidean n-space, sometimes called Cartesian space or simply n-space, is the space of all n-tuples of real numbers, (x_1, x_2, ..., x_n). Such n-tuples are sometimes called ...