$\def\C{\mathbb{C}}\def\CP{\mathbb{CP}}$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.
Best Answer
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.