Interactive Course on Projective Geometry

Appendix 3

Projective geometry is not only the study of lines or hyperplanes passing through the origin of a (finite-dimensional) vector space. In this chapter we study the manifolds (generalized surfaces) of k-dimensional subspaces passing through the origin. These are the ubiquitous Grassmann manifolds , which appear in differential geometry, topology, the theory of hypergeometric functions, the study of solitons, and the theory of singularities.

We shall write down explicit charts for the Grassmann manifolds and we shall give explicit embeddings into the exterior algebra of the vector space.