Can Vector Fields be Extended to Submanifolds?

[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?

 

[Hurkyl]

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

 

[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.

 

[Chris Hillman]

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

Let M [be a submanofld of] M'. Show that any vector field X on M can be extended to a vector field on M'.

...there is a counterexample:

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

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

 

[WWGD]

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

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?.

 

[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!

 

[WWGD]

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

Thanks.