In Example 1.2.22 of these notes you will find a description of the manifold structure of $\DeclareMathOperator{\Gr}{Gr}$ $\newcommand{\bC}{\mathbb{C}}$ $\Gr_n(V)$, where $V$ is a finite dimensional complex Hilbert space. The finite dimensionality is used only in computing the dimension. It is described as a submanifold in the real Hilbert space of (bounded) self-adjoint operators $V\to V$. As such it has an induced metric.

If you fix a orthonormal basis $(e_n)_{n\geq 1}$ of $V$, then you can produce a stratification of $\Gr_k(V)$ by Schubert cells of finite codimension. One can show that these define cohomology classes spanning the cohomology of $\Gr_k(V)$; Appendix A of this paper of Daniel Cibotaru explains how one can associate cohomology classes to strata under certain conditions that are satisfied in this case.

These strata also have a Morse theoretic description.

In Example 1.2.22 of Nicolaescu - Lectures on the Geometry of Manifolds you will find a description of the manifold structure of $\DeclareMathOperator{\Gr}{Gr}\newcommand{\bC}{\mathbb{C}}$$\Gr_n(V)$, where $V$ is a finite dimensional complex Hilbert space. The finite dimensionality is used only in computing the dimension. It is described as a submanifold in the real Hilbert space of (bounded) self-adjoint operators $V\to V$. As such it has an induced metric.

If you fix a orthonormal basis $(e_n)_{n\geq 1}$ of $V$, then you can produce a stratification of $\Gr_k(V)$ by Schubert cells of finite codimension. One can show that these define cohomology classes spanning the cohomology of $\Gr_k(V)$; Appendix A of Localization formulae in odd K-theory by Daniel Cibotaru explains how one can associate cohomology classes to strata under certain conditions that are satisfied in this case.

These strata also have a Morse theoretic description.