# The space of skew-symmetric orthogonal matrices

Let $$M_n \subseteq SO(2n)$$ be the set of real $$2n \times 2n$$ matrices $$J$$ satisfying $$J + J^{T} = 0$$ and $$J J^T = I$$. Equivalently, these are the linear transformations such that, for all $$x \in \mathbb{R}^{2n}$$, we have $$\langle Jx, Jx \rangle = \langle x, x \rangle$$ and $$\langle Jx, x \rangle = 0$$. They can also be viewed as the linear complex structures on $$\mathbb{R}^{2n}$$ which preserve the inner product.

I'd like to understand $$M_n$$ better as a topological space, namely an $$(n^2-n)$$-manifold.

$$M_1$$ is just a discrete space consisting of two matrices: the anticlockwise and clockwise rotations by $$\pi/2$$.

For $$n \geq 2$$, we can see that $$M_n$$ is an $$M_{n-1}$$-bundle over $$S^{2n-2}$$. Specifically, given an arbitrary unit vector $$x$$, the image $$y := Jx$$ must lie in the intersection $$S^{2n-2}$$ of the orthogonal complement of $$x$$ with the unit sphere $$S^{2n-1}$$. Then the orthogonal complement of the space spanned by $$x$$ and $$y$$ is isomorphic to $$\mathbb{R}^{2n-2}$$, and the restriction of $$J$$ to this space can be any element of $$M_{n-1}$$.

Since the even-dimensional spheres are all simply-connected, it follows (by induction) that $$M_n$$ has two connected components for all $$n \in \mathbb{N}$$, each of which is simply-connected. For instance, $$M_2$$ is the union of two disjoint 2-spheres: the left- and right-isoclinic rotations by $$\pi/2$$. The two connected components of $$M_n$$ are two conjugacy classes in $$SO(2n)$$; they are interchanged by conjugating with an arbitrary reflection in $$O(2n)$$.

Is [each connected component of] $$M_n$$ homeomorphic to a known well-studied space? They're each an:

$$S^2$$-bundle over an $$S^4$$-bundle over $$\dots$$ an $$S^{2n-4}$$-bundle over $$S^{2n-2}$$

but that's not really very much information; can we say anything more specific about their topology?

• Your $M_n$ is the Riemannian symmetric space $\mathrm{SO}(2n)/\mathrm{U}(n)$. I believe that its topology is quite well studied from that point of view. Sep 18 '20 at 17:43
• Thanks! Yes, it looks like the space is called DIII in Cartan's classification of compact Riemannian symmetric spaces. (I'll accept this if you post it as an answer.) Sep 18 '20 at 18:35
• yes, this is class DIII, the two connected components are distinguished by the Pfaffian, which equals $\pm 1$; in the physics context this matrix $M_n$ is the scattering matrix of a superconductor with preserved time-reversal symmetry but broken spin-rotation symmetry. The sign of the Pfaffian then distinguishes topologically trivial from topologically nontrivial superconductors. Sep 18 '20 at 19:30
• In Morse Theory, Milnor has proved that $\mathrm{O}(2n)/\mathrm{U}(n)$ can be identified with orthogonal linear transformations $J$ with $J^2=-I$ i.e. complex structure. Sep 18 '20 at 19:34

Your $$M_n$$ is (two copies of) the Riemannian symmetric space $$\mathrm{SO}(2n)/\mathrm{U}(n)$$ (which is $$DI\!I\!I$$ in Cartan's nomenclature). Its topology is well-studied from that point of view.