How is the dimension of a Grassmannian defined?

Asked Jan 19 '22 at 05:47

Active Jan 19 '22 at 05:51

Viewed 90 times

I'm new to topology, I'm struggling with how the dimension of a (real) Grassmannian Gr(k,n) is defined?

I know the definition of dimension of a vector space, but Gr(k,n) does not seem like a vector space (if it is what is the vector addition and over which field is it a vector space?)

In such a case, how is the dimension of Gr(k,n) defined (could you provide a definition)?

asked Jan 19 '22 at 05:47

Anon

2,460
1
15
23

$\operatorname{Gr}(k, n)$ is a smooth manifold. Do you know what a smooth manifold is? Such objects have a notion of dimension. – Michael Albanese Jan 19 '22 at 13:12
@MichaelAlbanese I know the definition of a manifold (Locally Euclidean, Hausdorff space). Could you briefly define the dimension of a smooth manifold as an answer. – Anon Jan 20 '22 at 01:33
1

It's just the dimension of the Euclidean space. – Michael Albanese Jan 20 '22 at 01:53

How is the dimension of a Grassmannian defined?

0 Answers0