Riemannian metric on a flag variety

Question

$\def\C{\mathbb{C}}\def\CP{\mathbb{CP}}$Every complex projective space $\CP^n$ has a natural Riemannian metric, the Fubini–Study metric, which is defined via the quotient definition of $\CP^n = \C^n/\C^* \cong S^{2n - 1}/S^1$ to be the metric descending from the ($S^1$-invariant) round metric on $S^{2n - 1}$. This is not the only metric one could define, but it is a nice one; for me, it is nice because I don't have to choose coordinates to define it.

My question is about the grassmannian varieties or, more generally, the flag varieties. Of course, one could define metrics on them by restriction along their Plücker embeddings in projective space, but this is a little indirect: one step removed from the end result for grassmannians, and two steps for finer flag varieties (which it seems to me would have to be embedded first into products of grassmannians).

If you agree upon the standard basis on $\C^n$, then you can fix a complex inner product $(,)$ on it and then define a metric on, say, the grassmannian $G(k, n)$ by specifying the real inner product on each tangent space $T_W$ to $W \subset \C^n$ via: $$(\phi, \psi) = \Re\operatorname{tr}(\psi^* \phi), \qquad \text{for } \phi, \psi \in T_W \cong \operatorname{Hom}(W, \C^n/W) \text{ with } \C^n/W \cong W^\perp \subset \C^n.$$ This has a nice natural feel other than the choice of basis on $\C^n$. In the special case where $k = 1$, I suppose one could check directly that this agrees with the Fubini–Study metric up to a scalar multiple; I suppose a nice abstract reason is that it is homogeneous (I am uneducated: does this uniquely-up-to-scalars determine it?)

Is there a good "intrinsic" way of defining a metric on $G(k,n)$ or more generally, any flag variety of (possibly partial) flags in $\C^n$, that generalizes the Fubini–Study metric?

Edit: I have streamlined the definition of the metric from the original version of the question. I'm satisfied with Will's answer now that it occurs to me that in fact, even the Fubini–Study metric requires a little non-naturality, in that to specify the isomorphism $\C^n/\C^* \cong S^{2n - 1}/S^1$, you need to be able to pick out the sphere in $\C^n$, which requires identifying the standard inner product (metric). So this really is no worse.

Answer 1

A truly coordinate-free metric would be preserved by the action of $GL_n(\mathbb C)$, which I think is impossible, for instance because it would produce an invariant measure, which should be impossible.

Consider the manifold of (ordered or unordered) sets of $k$ orthonormal vectors. This is the appropriate analogue to $S^{2n-1}$. The analogue to $S^1$ is $U^k$. We thus need to find a nice $U^k$-invariant metric on this space.

If $v_1,...,v_k$ is a set of orthonormal vectors, the tangent space consists of derivatives $dv_1,...,dv_k$ satisfying $v_i\cdot dv_i=0$, $v_i \cdot dv_j+v_j\cdot dv_i=0$. The obvious metric is just $\sum_{i=1}^k dv_i^a \cdot dv_i^b$, which is $U(n)$-invariant.

This is clearly equivalent to Fubini-Study when $k=1$. If I understand your metric correctly, it's equivalent to yours, since the sum should just be calculating the trace using a basis.

Answer 2

The Fubini-Study metric on $\mathbb P^n$ arises as the curvature of a line bundle. More precisely, once you fix a hermitian inner product in the complex vector space you are projectivizing, you get a natural metric on the tautological line bundle $\mathcal O(-1)$ on the projective space. If you take the dual metric on $\mathcal O(1)$ and you compute its Chern curvature you find the Fubini-Study metric.

Once you look at things in this way, on the Grassmannian you have a completely analogous construction (which, after all, gives you the induced metric of the Plücker embedding in a somewhat more direct way).

Call $S\to G(k,n)$ the tautological holomorphic vector bundle of rank $k$ (the fiber over each point is the $k$-dimensional subspace of the vector space $V$ you are considering). Then, once you are given a hermitian inner product on $V$, you can endow naturally $S$ with a smooth hermitian metric. In particular, you can induce a smooth hermitian metric on the holomorphic line bundle $\det S^*\to G(k,n)$. Now, compute its Chern curvature to discover that it is a (strictly) positive $(1,1)$-form, so that it gives you a hermitian metric on $G(k,n)$.

Observe that the map given by the complete linear system associated to $\det S^*$ is nothing more than the Plücker embedding, so that the metric we constructed here is what you were looking for, but given in a more intrinsic way.

Notice finally that this is really a generalization of the Fubini-Study metric, since when you look to $G(1,n)$, then $S=\mathcal O(-1)$ and the construction is as above.

There are of course more general constructions for complete or incomplete flag varieties, but the guiding philosophy is again the same: to put a natural hermitian metric on some (positive) line bundle and then to take the metric on your space to be its Chern curvature. For more details and computations you can look for instance at this paper by J.-P. Demailly.

Answer 3

Here's another rather easy way to define a natural Riemannian metric on a Grassmannian $G(k,n)$. Let $V$ be an $n$-dimensional complex vector space with a hermitian inner product. Given any $k$-dimensional subspace $L \subset V$, there is a natural isomorphism $$ T_LG(k,n) = \operatorname{Hom}(L, V/L) = L^* \otimes V/L $$ The hermitian inner product on $V$ induces a natural hermitian inner on $L^*\otimes V/L$. This defines a hermitian metric on $G(k,n)$ that is easily checked to be invariant under the action of $SU(n)$ on $G(k,n)$. What's less obvious from this construction is that the metric is K\"ahler.

This approach also works for flag manifolds.

I haven't looked, but I'm pretty sure this is all explained in Griffiths and Harris.

Answer 4

Let $G$ be a (semi)simple compact connected Lie group. Any flag variety $M=G^{\mathbb{C}}/P=G/K$, where $K=C(S)$ is the centralizer of a torus $S$ of $G$, admits a reductive decomposition $\frak{g}=\frak{k}\oplus\frak{m}$ with respect to the inner product $( \ , \ ) =-B( \ , \ )$ induced by the negative of the Killing form of $G$, that is

$\left[\frak {k}, \frak{ m}\right] \subset \frak{ m}$

Then, we identify ${\frak{m}} = T_{o} M$ with the tangent space $T_{o} M$ of $M$ at $o=eK$. Moreover, since the isotropy representation $\chi : K\to SO(\frak{m})$ is equivalent with the restriction of the adjoint representation $Ad\big|_{K}$ on the subaspace $\frak{m}\subset\frak{g}$, any $G$-invariant Riemannian metric on $M$, say $\rho=\rho( \ , \ )$ corresponds to an $Ad(K)$-invariant inner product on $\frak{m}$, say $\langle \ , \ \rangle$.

If we assume that $ {\frak m} $ admits a decomposition into mutually non-equivalent irreducible $\mbox{Ad}(K)$-modules as follows (such a decomposition always exists and is given in terms of $\frak{t}$-roots, see Alekseevsky's papers):

$ {\frak m} = {\frak m}_1 \oplus \cdots \oplus {\frak m}_q $

then a $G$-invariant metric on $G/K$ can be written as

$ \rho= \langle \ , \ \rangle = x_1 ( \ , \ )\big|_{\frak{m}_1} + \cdots + x_q ( \ , \ )\big|_{\frak{m}_q} $

for positive real numbers

$(x_1, \cdots, x_q)\in\mathbb{R}^{q}_{+}$

(such metrics are usually called diagonal). Thus, the space of $G$-invariant metrics on flag manifolds can be parametrizided by real positive nymbers. Note that since the $Ad(K)$-submodules $\frak{m}_{i}$ are always pairwise inequivalent, any $G$-invariant metric has the above form. In particular, any $G$-invariant symmetric covariant 2-tensors on $G/K$ are of the same form as the Riemannian metrics (although they are not necessarilly positive definite).