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

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    $\begingroup$ $U(H)$ (unitary group of a Separable Hilbert space) is a pretty nice contractible Frechet manifold. As a subgroup, $U(n)$ acts freely on it, so quotient would be my first candidate for a differential-geometric model of BU(n) $\endgroup$
    – Denis T
    Commented Nov 13, 2021 at 20:45
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    $\begingroup$ The title asks a slightly different question than the actual post. A “better” classifying object for the unitary group is the classifying stack B(U(n)), and it is certainly possible to talk about the differential geometry of B(U(n)). For example, Freed and Hopkins in “Chern–Weil forms and abstract homotopy theory” compute the de Rham complex of B(U(n)), which turns out to be precisely isomorphic (and not just quasi-isomorphic) to the algebra of Chern classes. $\endgroup$
    – Dmitri Pavlov
    Commented Nov 14, 2021 at 3:24
  • $\begingroup$ You might find this paper interesting: arxiv.org/abs/1301.5959 ("Chern-Weil forms and abstract homotopy theory", by Dan Freed and Mike Hopkins) $\endgroup$
    – André Henriques
    Commented Nov 19, 2021 at 17:15

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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.

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  • $\begingroup$ Thank you for the reference, these are very nice notes! Does this give a "curvature" description of the classifying space? I put curvature in "..." since I'm not sure how one makes sense of the curvature of an infinite-dimensional manifold. Essentially, I want to know whether the classifying space is positively or negatively curved (in any sense of the word). $\endgroup$
    – Kaleb
    Commented Nov 16, 2021 at 8:00
  • $\begingroup$ I don't know how to define Ricci or scalar curvature since they involve infinite summations. $\endgroup$
    – Liviu Nicolaescu
    Commented Nov 16, 2021 at 11:01
  • $\begingroup$ MathJax warning: $\DeclareMathOperator{\Gr}{Gr}$ $\newcommand{\bC}{\mathbb{C}}$ $\Gr_n(V)$ would be fine TeX, but, in MathJax, forces unwanted spaces. As ugly as it is, you must remove all spaces: $\DeclareMathOperator{\Gr}{Gr}\newcommand{\bC}{\mathbb{C}}$$\Gr_n(V)$. I have edited accordingly. $\endgroup$
    – LSpice
    Commented Jan 12, 2022 at 20:06
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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.

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