# The Grassmann Manifold

• Shiing-shen Chern
Part of the Universitext book series (UTX)

## Abstract

Let
$${C_{N + 1}} = Cx \ldots xC,\;N + 1\,factors$$
(8.1)
be the complex number space of N + 1 dimensions. Let GL(N+1, C) be the general linear group in N + 1 complex variables, which we identify with the group of all (N+1) × (N+1) non-singular matrices with complex elements. Suppose GL(n+l,C) acts on CN+1 to the right, as described by
$$\left( {{z^0}, \ldots ,{z^N}} \right) \to \left( {{z^0}, \ldots ,{z^N}} \right)g,\;gGL\left( {n + l,C} \right)$$
(8.2)
Among the subgroups of GL(N+1, C) are: (1) the unitary group U(N+1), which consists of all matrices g satisfying
$${t_{g\bar g}} = I$$
(8.3)
where I is the identity matrix; (2) the group GL(k+1, N-k, C), consisting of all non-singular matrices of the form
$$\left( {\underbrace {\begin{array}{*{20}{c}} x \\ x \\\end{array}}_{k + 1}\,\underbrace {\begin{array}{*{20}{c}} 0 \\ x \\\end{array}}_{N - k}} \right)\,\begin{array}{*{20}{c}} {\} \,k + 1} \\ {\} \,N - k} \\\end{array}$$
(8.4)
where the elements at the upper-right corner are zero. The group GL(k+l, N-k, C) is the subgroup of all elements of GL(N+1, C) leaving fixed the (k+1)-dimensional subspace of CN+1 spanned by the first k+1 coordinate vectors.

## Keywords

Meromorphic Function Chern Class Fiber Dimension Schubert Variety Hermitian Structure
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.