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 $\text{Gr}_n(\mathbb{C}^{\infty})$?