Extending Vector Fields.

1. Nov 20, 2007

WWGD

Hi: I am going over Lee's Riemm. mflds, and there is an exercise that asks:

Let M<M' (< is subset) be an embedded submanifold.

Show that any vector field X on M can be extended to a vector field on M'.

Now, I don't know if he means that X can be extended to the _whole_ of

M', because otherwise, there is a counterexample:

dt/t on (0,1) as a subset of IR cannot be extended to the whole of IR.

Anyone know?.

What Would Gauss Do?

2. Nov 20, 2007

Hurkyl

Staff Emeritus
Are you sure M is not supposed to be a closed submanifold?

3. Nov 21, 2007

WWGD

Hi. I did not see this stated, and I cannot see if it is being assumed somehow.

Only conditions I saw where that M<M' , and M embedded submanifold of M',

and a V.Field is defined in M.

Re the second issue, of extensions, I guess these extensions are local, tho

not necessarily global, right?. The answer would seem to be yes pretty

clearly, but my intuition has failed me before.

4. Nov 21, 2007

Chris Hillman

Consider the function with constant value one on (0,1). How many ways can you extend it to R?

5. Nov 23, 2007

WWGD

Thanks, Chris, I am not sure I get the hint; there are uncountably many ways,

for a continuous extension, most obvious extension being f==1 in IR (attach

a line segment of slope m, m in (-oo,oo) , maybe

smaller cardinality for smooth extensions . For smoothness I would imagine

some combination of e^-x 's attached to both ends, or maybe some other

bump functions ( Lee does not specify if X is C^1 , or C^k, C^oo).

All I can think of when I think of immersed submanifolds is slice coordinates, tho

this does not seem to make sense for 1-manifolds like (0,1) in IR.

Am I on the right track?.

6. Nov 23, 2007

Chris Hillman

Mrmph... never mind the hint, I was looking at the wrong textbook in Lee's excellent trilogy, my mistake! Unfortunately you are using the one I don't have, but can you state the what exercise you are attempting? I might be able to obtain a copy next week. I expect we will be able to figure it out!

Last edited: Nov 23, 2007
7. Nov 23, 2007

WWGD

Yes, this is Lee's Riemannian mflds, problem 2.3, part b, p.15 in my edition.

Thanks.