Classifying space $\text{BU}(n)$ from the differential-geometric point of view? The classifying space of a topological group $G$ is the quotient of $EG$ (a topological space with vanishing homotopy groups) by a proper free action of $G$. The standard notation for the classifying space of $G$ is $BG$, with $B$ short for "classifying space".
The classifying space of the unitary group $\text{U}(n)$ is $\text{BU}(n) \simeq \text{Gr}_n(\mathbb{C}^{\infty})$, where $\text{Gr}_n(\mathbb{C}^{\infty})$ is the "inductive-limit" of the embeddings $$\text{Gr}_n(\mathbb{C}^k) \hookrightarrow \text{Gr}_n(\mathbb{C}^{k+1}) \hookrightarrow \cdots.$$
The infinite Grassmannian $\text{Gr}_n(\mathbb{C}^{\infty})$ is not finite-dimensional, and is therefore not a manifold in the usual sense.

Are there any results concerning the differential-geometric structure of $\text{Gr}_n(\mathbb{C}^{\infty})$?


Can we speak of the curvature of $\text{Gr}_n(\mathbb{C}^{\infty})$? Do we know anything about the curvature of "Riemannian metrics" which reside on $\text{Gr}_n(\mathbb{C}^{\infty})$?

 A: 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.
A: One prerequisite to any differential-geometric point of view of course is that $G$ is a Lie group rather than just a topological group.
Then, one option is to pass from manifolds to Lie groupoids:

*

*If $G$ is a Lie group, then there is a Lie groupoid with one object whose automorphism group is $G$; let's denote this Lie groupoid by $BG$.


*If $M$ is a smooth manifold, then there is a Lie groupoid with objects $M$ and only identity morphisms. Let's denote this Lie groupoid by $M$ again.
"The correct" kind of morphisms between Lie groupoids are "smooth anafunctors" a.k.a. "principal bibundles". All smooth anafunctors between two Lie groupoids from a category (Lie groupoids become a bicategory when equipped with such morphisms).
There is a canonical equivalence of categories:
$$
\operatorname{Hom}(M,BG) \cong \operatorname{Bun}_G(M).
$$
This is the Lie-groupoidal version for the classification of $G$-bundles by morphisms to a fixed object, $BG$. Note that everything takes places in finite-dimensional smooth manifolds, and it works for any Lie group $G$.
The relation to the classical picture is established by geometric realization (of topological groupoids). For example, $\lvert BG\rvert$ is a model for the classifying space, and smooth anafunctors induce homotopy classes of continuous maps.
This pov is explained in Section 2 of:
Nikolaus, Thomas; Waldorf, Konrad, Four equivalent versions of nonabelian gerbes, Pac. J. Math. 264, No. 2, 355-420 (2013). ZBL1286.55006.
