ГЕОМЕТРИЯ МНОГООБРАЗИЯ ГРАССМАНА

The work investigates the differential-geometric structure of the Grassmannian manifold Gr(k, n),
consisting of all k-dimensional linear subspaces in an n-dimensional real vector space. The primary
focus is on the explicit construction of a smooth atlas and the analysis of the transition functions
between its charts.
It is proven that for an arbitrary set of indices I = {i 1 < ··· < i k }, the corresponding coordinate map
ϕ I : U I → R k(n−k) is a homeomorphism, where U I is an open subset of Gr(k, n). The smoothness
(infinite differentiability) of the transition functions ψ IJ = ϕ J ◦ϕ −1
I
on the intersections of the charts
is established.
Using the example of the manifold Gr(2, 4), detailed calculations are performed to demonstrate
the explicit form of the transition functions between different coordinate charts. It is shown that
these functions are expressed as rational mappings with non-zero denominators, which ensures their
correctness and smoothness.
The obtained results are of significant importance for applications in differential geometry, algebraic
topology, and mathematical physics, where Grassmannian manifolds play a key role.