Variation of normals along loops

Question

I've constructed a quotient space $M/\sim$ in $\mathbb{R}^d$ that must be a $2$-manifold. If $M/ \sim$ is a sphere then I know that its normal spaces must vary at least 90 degrees. That is $M / \sim$ must exhibit a full sphere of normal spaces. Suppose instead that $M / \sim$ has a handle. Can I always choose a loop around the handle that also implies that the normals of $M / \sim$ must vary by at least 90 degrees ?

Answer 1

Edit: The Clifford torus (product of two circles) in $\mathbb{R}^4$ has trivial normal and trivial tangent bundle. Its normal space is parallel, but varies by 90 degrees, so it is not a counterexample.

If the normal spaces all live within a right angle of one of them, then orthogonal projection to that one trivializes the normal bundle. Since the sum of tangent and normal bundle is trivial, the characteristic classes of the tangent bundle are trivial, and the surface is a torus or Klein bottle. So only the torus or Klein bottle can have normal bundle in $\mathbb{R}^d$ with no two perpendicular normal spaces. But I don't know when that happens.

Edit: A particular vector $v \in \mathbb{R}^d$ is normal to a compact submanifold $M \subset \mathbb{R}^d$ without boundary just where the linear function $f \colon x \in M \mapsto v \cdot x \in \mathbb{R}$ is critical on $M$. But every continuous function has a maximum and a minimum. So every vector belongs to the normal space at some point. So the normal spaces at different points must contain perpendicular vectors.