An integral characterization of Euclidean space

The Euclidean Functional Integral

We give a mathematical construction of Euclidean quantum field theory on certain curved backgrounds. We focus on generalizing Osterwalder Schrader quantization, as these methods have proved useful to establish estimates for interacting fields on flat space-times. In this picture, a static Killing vector generates translations in Euclidean time, and physical positivity is played by positivity un...

Characterizations of Slant Ruled Surfaces in the Euclidean 3-space

In this study, we give the relationships between the conical curvatures of ruled surfaces generated by the unit vectors of the ruling, central normal and central tangent of a ruled surface in the Euclidean 3-space E^3. We obtain differential equations characterizing slant ruled surfaces and if the reference ruled surface is a slant ruled surface, we give the conditions for the surfaces generate...

Structure and characterization of ruled surfaces in Euclidean 3-space

Keywords: Ruled surface Structure function Pitch function Angle function of pitch Weingarten surface Binormal ruled surface a b s t r a c t In this paper, using the elementary method we study ruled surfaces, the simplest foliated submanifolds, in Euclidean 3-space. We define structure functions of the ruled surfaces, the invariants of non-developable ruled surfaces and discuss geometric propert...

The space of clouds in an Euclidean space

We study the space Nm d of clouds in Rd (ordered sets of m points modulo the action of the group of affine isometries). We show that Nm d is a smooth space, stratified over a certain hyperplane arrangement in Rm. We give an algorithm to list all the chambers and other strata (this is independent of d). With the help of a computer, we obtain the list of all the chambers for m ≤ 9 and all the str...

A nonstandard analysis characterization of submanifolds in Euclidean space