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Let M be a surface in $R^3$. $E_1, E_2, E_3$ are orthonormal vector fields defined on M.

I wonder how one can extend say $E_1$ to some open set of $R^3$.

Naturally one wants to move $E_1$ along the normal vector of M at each point. However, those normal vectors at different points may intersect making the new extended vector fields ill defined.

How is such extension always possible?

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  • $\begingroup$ en.wikipedia.org/wiki/Tubular_neighborhood does that help? the point is that going a very small (fixed) amount in the normal direction should be a 1-1 map. $\endgroup$ Oct 25, 2013 at 22:04
  • $\begingroup$ Does this amount depend on the point of the surface? Is there a same positive amount for all points on the surface? $\endgroup$
    – noot
    Oct 25, 2013 at 22:38
  • $\begingroup$ if your surface is compact, then you can of course find a lower bound independent of the surface $\endgroup$ Oct 26, 2013 at 5:08

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It is not always possible, for example you cannot extend the unit normal vector to the 2-sphere to a nonzero vector field inside the ball. I gave this problem on my 3rd year undergraduate analysis examination last year, and I think all of the students were able to do it. Essentially it is the Brouwer fixed point theorem. You map the closed ball to the sphere, by the value of the unit vector field at each point. The map restricts to the identity on the boundary. It then pulls back the spherical angle form, defined away from the origin, to a closed 2-form on the ball, with integral $4 \pi$ over the sphere. But such a form must be exact on the ball, by the Poincare lemma (triviality of the deRham cohomology of the ball) so has integral $0$ over the sphere.

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  • $\begingroup$ Such extension was taken for granted in the textbook 'Elementary differential geometry' by O'Neill. It was crucial for deriving the fundamental equations in section 6.1. Can anyone take a look? $\endgroup$
    – noot
    Oct 25, 2013 at 21:38
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    $\begingroup$ it seems that there is some confusion between extending the vector fields "some open set" and "a given open set." the first is possible, the second is not, as this answer shows $\endgroup$ Oct 25, 2013 at 21:49
  • $\begingroup$ how can one extend the vector field to some open set? $\endgroup$
    – noot
    Oct 25, 2013 at 21:52
  • $\begingroup$ You can't always extend locally either, if your surface is immersed but not embedded. If the surface is embedded, you use a partition of unity argument to reduce the problem to a local question, and then locally the surface in its ambient space can be identified with a flat plane in 3-dimensional space, so you can locally extend a vector field easily. You then piece together the local extensions, and then use Gram-Schimdt to make the extensions orthonormal. $\endgroup$
    – Ben McKay
    Oct 28, 2013 at 9:24

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