This is actually a series of questions posed by Guram Berishvili about the structure he calls marao. Everything I am going to write here I took (and messed up) from his home page which is all in Georgian. "Marao" (spelling is "March Dao" with "ch D" omitted) means "fan" in Georgian, but since this term is used to name an entirely unrelated structure in a very closely related topic, I decided to leave it untranslated.
So, a $d$-marao in a vector space $V$ is a set $\mathbf M$ of $d$-dimensional sub(-vector )spaces of $V$ with $V=\bigcup_{L\in\mathbf M}L$ and $L_1\cap L_2=\{0\}$ for distinct $L_1,L_2\in\mathbf M$. That is, every nonzero element of $V$ belongs to a unique $L\in\mathbf M$.
Most of his interest is in middle maraos - those with $\dim(V)=2d$ - but he also has some questions about general ones.
The motivation for considering maraos was to have a simple view of Hopf bundles. Note that for each $d$-marao $\mathbf M$, sending a line $\ell\subset V$ to the unique $L\in\mathbf M$ with $\ell\subset L$ gives a map $\pi_{\mathbf M}:\mathbf P(V)\to\mathbf M$, with $\pi_{\mathbf M}^{-1}(L)=\mathbf P(L)$ for $L\in\mathbf M$, so that each such $\mathbf M$ is the base of a bundle with ($\dim(V)-1$)-dimensional projective space as the total space and ($d-1$)-dimensional projective spaces as fibres.
Except that, although in view of this map $\mathbf M$ "morally" looks like a ($\dim(V)-d$)-dimensional homogeneous manifold, there is in fact no obvious structure on this set, and in fact most of Berishvili's questions are precisely around this. Specifically, here is sort of a central question.
For any $d$-dimensional division algebra $A$, a $d$-marao in $A^k$ (considered as a $kd$-dimensional vector space over the ground field) is given by the ($k-1$)-dimensional projective space over $A$. Straining this a bit, one might probably construct some maraos from more exotic objects like Rozenfeld projective planes. There might be also some examples coming from Severi-Brauer varieties. How exactly can one construct them? And are there still other examples?
For middle maraos Berishvili actually comes quite close to constructing such an algebra. The crucial property of middle maraos is that every distinct $L_1,L_2\in\mathbf M$ come with a canonical isomorphism $V\cong L_1\oplus L_2$, and the affine subspaces $L_1+x_1$ and $L_2+x_2$ intersect in a unique point for any $x_1,x_2\in V$. This in particular allows to view any $L\in\mathbf M$ as the graph of a linear map $$ (*)\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad _{L_1}L_{L_2}:L_1\to L_2\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\phantom{(*)} $$ which is zero for $L=L_1$, undefined for $L=L_2$, and an isomorphism for all other $L$.
Note also that for any $L\in\mathbf M$ in a middle marao $\mathbf M$, any $o\in V\setminus L$ determines a bijection $$ L\leftrightarrow(\mathbf M\setminus\{L\}) $$ as follows: to $x\in L$ corresponds the unique $L_o(x)\in\mathbf M\setminus\{L\}$ with $x+o\in L_o(x)$ and to $L'\in\mathbf M\setminus\{L\}$ corresponds the unique $o_L(L')\in L\cap(L'-o)$.
This can be viewed as some kind of an atlas to endow $\mathbf M$ with the structure of some kind of variety, and shows that it is some sort of analog of a sphere for general fields (a vector space with a point added).
But it also can be used to define a multiplication on $L$. Let $O=L_o(0)$ (the unique element of $\mathbf M$ with $o\in O$). For every $x\in L$ we may, as in $(*)$ above, view $L_o(x)\in\mathbf M\setminus\{L\}$ as a linear map ${}_OL_o(x)_L:O\to L$. Denote this map by $o_x$; it satisfies $o_x(o)=x$ and is an isomorphism except for $x=0$, when it is the zero map. We can then pick any nonzero $e\in L$, and define a multiplication by $$ x*y:=o_xo_e^{-1}(y) $$ for any $y\in L$. By construction this multiplication is left distributive, $e$ is a two-sided unit, and given any two nonzero elements out of $x,y,z$ there is a unique third one with $x*y=z$.
In terms of this multiplication, the above atlas is expressed as follows. $\mathbf M$ is obtained by gluing two copies $L^+$, $L^-$ of $L$ along the bijection $(\_)^{*(-1)}:L^+\setminus\{0\}\to L^-\setminus\{0\}$ sending an element to its inverse with respect to the multiplication.
Is the above multiplication also always right distributive?
There is some work on left distributive right quadratic multiplications related to homotopy groups of spheres, see e. g. "Square Rings Associated to Elements in Homotopy Groups of Spheres" by Baues and Iwase and some further references there. However in all examples I've been able to extract from there either the corresponding addition is noncommutative, or it seems essential to work over integers. Anyway, I've been unable to produce any maraos from that.
Finally, a couple of general questions.
Does $GL(V)$ act transitively on the set of all $d$-maraos in $V$?
For a given marao $\mathbf M$, does the subgroup of $GL(V)$ preserving $\mathbf M$ act transitively on $\mathbf M$?
In case it turns out that there are really wild examples, one may restrict the questions as follows. Obviously each $d$-marao $\mathbf M$ is a subset in the Grassmanian of $d$-dimensional subspaces of $V$. In fact one can even speak about the tangent space $T_L(\mathbf M)$ to $\mathbf M$ at $L\in\mathbf M$ as certain naturally definable subspace of the tangent space of the Grassmanian at $L$, the latter canonically identifiable with $\operatorname{Hom}(L,V/L)$; for middle maraos this subspace actually turns out to lie (entirely except for zero) inside the subset $\operatorname{Iso}(L,V/L)\subset\operatorname{Hom}(L,V/L)$ of isomorphisms.
One may then impose restrictions like $\mathbf M$ being a subvariety of the Grassmanian, or a closed subset (in one of several possible topologies). Then the final (imprecise) question is
Are such restrictions redundant? Do there exist maraos which are really "bad" subsets of the Grassmanian?
Slightly later -
Hope to study and refer properly to the work of Ben McKay he mentions in his answer and comments.
Presumably spheroids representing nontrivial elements of higher homotopy groups of various Grassmanians provide some nontrivial examples.
Exotic spheres might be also a source of interesting examples.
Also, it occurred to me that there might be some combinatorial examples coming from matroids (matroid Grassmanians, etc.)
Oh yes, and also probably the simplest question about maraos:
Does $d$ always divide $\dim(V)$?
Even over finite fields this is not entirely trivial (but true): it follows from the fact that $q^d-1$ divides $q^n-1$ only when $d$ divides $n$.
Still further update -
Among the references to the papers linked to in the answer by Ben McKay is Great circle fibrations of the three-sphere by Gluck and Warner with examples of middle maraos giving rise to nonsmooth and even disconnected subsets of the Grassmanian. This is for $d=2$ over reals, but it is more or less clear that in all other cases something similar might happen.
So in principle I should accept that answer, except that
- the very last question (about divisibility) seems to remain open;
- one might require $\mathbf M$ to be an algebraic subvariety of the Grassmanian and see what happens then.
Regarding this last one - one result by Yang in "Smooth great circle fibrations and an application to the topological Blaschke conjecture" seems to imply that over reals smooth 2-maraos in an even-dimensional space (except for $\dim=6$) are "smoothly linearizable". Maybe this might be translated into something algebro-geometric, I don't know.
So maybe I'll wait some more time...
