# Which vector bundles are induced by maximal disjoint families of subspaces?

In a sense this is a followup of my earlier question Are there nonlinear projective spaces?. In any case, if there is anything interesting about this question it is all by virtue of Guram Berishvili.

Fix a natural number $$n$$. Moreover fix your favorite field and category to work in - algebraic, complex, real, analytic, smooth, topological, whatever.

For a vector space $$V$$ let Gr$$_n(V)$$ be the Grassmanian of all $$n$$-dimensional subspaces of $$V$$.

Every morphism $$f:X\to$$ Gr$$_n(V)$$ gives rise to a vector bundle on $$X$$. It also gives rise to a family $$(V_x)_{x\in X}$$ of $$n$$-dimensional subspaces of $$V$$.

Let us impose on this family the condition that for any nonzero $$v\in V$$ there is a unique $$x\in X$$ with $$v\in V_x$$.

Call a vector bundle "special" (just to call it something) if it can be obtained from some morphism $$f$$ (for some $$V$$) with the property that the corresponding family $$(V_x)_{x\in X}$$ satisfies the above condition.

The only examples of "special" vector bundles that I know are related to that question I link to.

Let $$k\subset K$$ be a finite field extension of degree $$n$$. Then, for $$V$$ a $$K$$-vector space, we have embedding

P$$(V)\hookrightarrow$$ Gr$$_n(V_k)$$

(view $$K$$-lines in $$V$$ as $$n$$-dimensional $$k$$-subspaces of the $$k$$-vector space $$V$$). The corresponding family is the tautological one, and the $$k$$-vector bundle on P$$(V)$$ so obtained is the tautological $$K$$-line bundle viewed as an $$n$$-dimensional $$k$$-bundle.

More generally, $$K$$ might be a non-commutative (associative) division ring, or a (non-associative) octonion algebra (but then dimension must be restricted).

Are there any other examples?

Does for every $$X$$ exist a vector bundle on $$X$$ that is "special" in this sense?

In particular, which manifolds have the property that their tangent bundles are "special"?

For $$n=1$$, and only imposing uniqueness of $$x$$ with $$v\in V_x$$ without its existence, this more or less amounts to very ample line bundles. However for $$n>1$$ it is essentially more than requiring $$f$$ to be an embedding into the Grassmanian, not only because of existence, but also because there is additional restriction that all $$V_x$$ are not only pairwise different but also must have pairwise zero intersections.