## Motivation

We want a consistent way of perturbing a submanifold away from itself. For $0$-dimensional submanifolds, this is the same data as a nowhere-vanishing vector field: we may flow the points off themselves. For $r>0$, a nowhere-vanishing vector field may flow a submanifold along itself, which would be a disaster. The setup below takes this into account, exploiting the extra data we have at each point. For example, notwithstanding the Hairy Ball Theorem, the 2-sphere can still flow oriented 1-dimensional submanifolds (see Examples section below).

## A Precise Question

Let $M$ be an oriented, closed, connected Riemannian manifold of dimension $n$. Define $Gr_+(TM,r)$, the ``universal oriented Grassmannian bundle'': the fiber over a point $m$ is the space of oriented $r$-subspaces of the tangent space $T_mM$. Points on $Gr_+(TM,r)$ can be written $(m,\alpha)$ where $\alpha \in \Omega^r_mM$ is a non-zero wedge product of tangent vectors at $m$ considered up to positive rescaling.

The space $Gr_+(TM,r)$ comes with a natural vector bundle of rank $n-r$: the fiber over the point $(m,\alpha)$ is the oriented subspace of $T_mM$ determined by $\star \alpha$, the ``oriented orthogonal complement'' of the oriented subspace determined by $\alpha$. (Here the Hodge star $\star$ is taken with respect to the chosen orientation on $M$)

Let $Gr_+^{\perp}(TM,r)$ denote this vector bundle, the ``oriented orthogonal complement of the universal oriented Grassmannian bundle.'' This is a rank $n-r$ vector bundle on $Gr_+(TM,r)$.

When does $Gr_+^{\perp}(TM,r)$ admit a nowhere-vanishing global section?

## Examples

When $r=0$, $Gr_+^{\perp}(TM,0)$ has a nowhere-vanishing global section exactly when the Euler characteristic of $M$ is zero: Does a connected manifold with vanishing Euler characteristic admit a nowhere-vanishing vector field?

The case $Gr_+^{\perp}(S^2, 1)$ is fun. Given a point on $S^2$ (considered as a unit vector in $\mathbb{R}^3$) and a tangent vector at that point (considered as another unit vector othogonal to the first) we are trying to determine a unit vector at the point which is orthogonal to the tangent vector. In other words, given two unit vectors in $\mathbb{R}^3$ we need to continuously determine a third unit vector that is perpendicular to the first two. We may use the usual cross product in $\mathbb{R}^3$! A similar construction applies to $Gr_+^{\perp}(S^6, 1)$; see http://en.wikipedia.org/wiki/Seven-dimensional_cross_product .