Isometry group of projective space pdf

Given a metric space loosely, a set and a scheme for assigning distances between elements of the set, an isometry is a transformation which maps elements to the same or another metric space such that the distance between the image elements in the new metric space is equal to the distance between the elements in the original metric space. Examples lines are hyperplanes of p2 and they form a projective space of dimension 2. For the standard round metric, this has sectional curvature identically 1. A manifolds, fundamental group and covering spaces. A discrete isometry group is an isometry group such that for every point of the space the set of images of the point under the isometries is a discrete set. G leaves m invariant in the projective space and g is the isometry group of m. We define the automorphism group or isometry group of our positive definite even lattice l as follows. A metric introduced on a projective space yields a homogeneous metric space known as a cayleyklein geometry. A key to the projective model of homogeneous metric spaces andrey sokolov school of physics, university of melbourne, parkville, vic 3010, australia email. Algebraic reflexivity of the isometry group of some spaces. He describes the isometry group of compact riemannian naturally reductive spaces. A key to the projective model of homogeneous metric spaces.

If the answer depends on the metric choices, then give the isommetry for your metric choices. They preserve lengths and angles however while direct isometries. The group of affine transformations is a subgroup of the previous one. Moreover, it is a nontrivial fact that this group is a lie group and therefore has a manifold structure and a dimension. We will now be interested by degenerate model spaces. In pseudoeuclidean space the metric is replaced with an isotropic quadratic form. In this chapter, formal definitions and properties of projective spaces are given. The elements of the projective special linear group psl2,r are. The fundamental theorem of projective geometry if a fieldf has no nontrivial. We study isometry groups of the simply connected space forms. The isometry group of outer space connecting repositories. Euclidean geometry is hierarchically structured by groups of point transformations. Request pdf algebraic reflexivity of the isometry group of some spaces of lipschitz functions we show that the isometry groups of lipx,d and lipx,d. The projective line is useful to introduce projective notions, such as the crossratio, in a simple and intuitive way.

There is an isometric copy h4 q of hyperbolic space with x 0 roydens theorem. Isometry group of a homogeneous space mathoverflow. On discrete projective transformation groups of riemannian manifolds. It is the completion of the ane line with a particular projective point, the point at in nity, as will be further detailed in this chapter. The general group, which transforms any straight line and any plane into another straight line or, correspondingly, another plane, is the group of projective transformations. Isometry groups of hyperbolic space ilyas khan abstract. A characterization of complex projective space up to. Can someone explain with some simple examples what is meant by the isometry and isometry group. In euclidean space, r n, it is easy to see that, for all v2r, the translation map t v. The cullervogtmann outer space, cv n, is the analogue of teichmuller space for out f n and is a space of metric graphs with fundamental group of rank n as for teichmuller space, one can define the lipschitz metric of cv n with a resulting metric which is not symmetric. Geometry of a complex projective space from the viewpoint of its. Im a student of physics and i often come across this term in texts of general relativity for various. Real projective space admits a constant positive scalar curvature metric, coming from the double cover by the standard round sphere the antipodal map is locally an isometry. Simple lie algebras are a nice example of this phenomenon.