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On $S^{1}$ there are three choices for a vector pointed inward to the same angle at the tangent line at every point. The vector is either in the opposite direction as the normal, or to the left of it, or to the right of it. (clockwise, counterclockwise, and all inward)

I can skip $S^{2}$ because of the hairy ball theorem. What about $S^{3}$ and $S^{7}$? I know that they're both uncombable.

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    @deoxygerbe: every vector field on $S^k$ (as a embedded submanifold of $R^{k+1}$ can be written as $v = v_n N + v_p$ where $v_n$ is the normal component to $S^k$ and $v_p$ is a vector field along $S^k$ that is tangent to it. By rescaling $v_p$ (multiplying it by a function), you see that your question is **equivalent** to classifying all nonvanishing vector fields _along_ (tangent to) the spheres.2012-03-08

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I think what you want is this: 1) write points on the circle as $z=(x,y)$ or $x+iy$, working in the algebra of complex numbers, then the unit inward normal to the circle at point $z$ is $N=-z$, and the various (constant length) vectors pointing inward at constant angle to $N$ are $N+aiz=(-x-ay,-y+ax)$ for any real constant $a$, or positive multiples of that. What you called right and left of the inner normal correspond to $a$ positive and negative respectively.

2) Write points on the 3-sphere as $q=(w,x,y,z)$ or $w+ix+jy+kz$, working in the algebra of quaternions, then the unit inward normal to the 3-sphere at point $q$ is $N=-q$, and the constant length vectors pointing inward at constant angle to $N$ are $N+(ai+bj+ck)q$ for any real constants $a$, $b$, $c$, or positive multiples of that.

3) For the 7-sphere use the same type of construction with the octonian algebra.