Another proof that a vector field on the sphere must have a zero?

[lavinia]

The proofs I have seen that a vector field on the 2-sphere must have a zero rely on the general theorem that the index of any vector field on a manifold equals the manifold's Euler characteristic.

How about this for a proof that does not appeal to this general theorem?

The tangent circle bundle of the 2 sphere is homeomorphic to real projective 3 space.
A non-zero vector field would be a map from

[tex] S^2 -> RP^3 [/tex]

which has a left inverse, p, where p is just the bundle projection map.

Otherwise put,

p o v = identity on [tex] S^2 [/tex]

But the second real homology of projective 3 space is zero so p o v must equal zero on the fundamental cycle of the 2 sphere. This contradicts the equation p o v = identity.

 

[Landau]

I learned this as an easy application of homology theory (used to define the degree of a map S^n->S^n):

Theorem: If f,g:S^n->S^n are continuous maps such that f(x)=/g(x) for all x, then f is homotopic equivalent to -g (i.e. the composition of g and the antipodal map).

Corollary: If f is as above with n even, then there exists x such that f(x)=x or f(x)=-x.
(proof: otherwise f would be homotopic equivalent to both the identity and the antipodal map, so would have degree both equal to 1 and (-1)^{n+1}=-1)

Corollary: Any vector field v on a 2k-sphere must have a zero.
Proof: otherwise v/|v| does not take any x to either x or -x, in contradiction to the previous result.

 

[lavinia]

I learned this as an easy application of homology theory (used to define the degree of a map S^n->S^n):

Theorem: If f,g:S^n->S^n are continuous maps such that f(x)=/g(x) for all x, then f is homotopic equivalent to -g (i.e. the composition of g and the antipodal map).

Corollary: If f is as above with n even, then there exists x such that f(x)=x or f(x)=-x.
(proof: otherwise f would be homotopic equivalent to both the identity and the antipodal map, so would have degree both equal to 1 and (-1)^{n+1}=-1)

Corollary: Any vector field v on a 2k-sphere must have a zero.
Proof: otherwise v/|v| does not take any x to either x or -x, in contradiction to the previous result.

Right. I know this proof. It is lovely.

 

[quasar987]

Fun proof lavinia, but could you tell me why the tangent cicle bundle of S^2 is homeomorphic to RP^3?

Thx

 

[mathwonk]

my favorite is lefschetz's proof. chose any point on the sphere where the vector is not zero. remove a small disk about that point. a bit of thinking will convince you that the degree of the map defined by the vector field on the boundary of the complementary disc is 2, hence on the complement of the disc around that point, there are 2 zeroes counting multiplicities. this takes a little imagination to visualize it but it is very elementary.

 

[lavinia]

Fun proof lavinia, but could you tell me why the tangent cicle bundle of S^2 is homeomorphic to RP^3?

Thx

SO(3) acts on the sphere by rotations. The derivatives of these rotations define a smooth action of SO(3) on the tangent circle bundle of the 2 sphere. This action transitive and is without fixed points. Thus the tangent circle bundle of the 2 sphere is diffeomorphic to SO(3).

 

[quasar987]

I see, thanks.

 

[lavinia]

I see, thanks.

Another proof which I would like to try - maybe we can do it together - would be to split the tangent circle bundle along the equator of the 2 sphere into two solid tori then study the attaching map. This could be figured out from the coordinate transformation z -> 1/z.

I think we would be looking to see that twice the homology class of one of the circles on one bounding torus is sent the identity loop in the other. This would mean that the fundamental group of the tangent circle bundle contains a copy of Z/2Z and that would mean that there could not be a non-zero vector field.

 

[quasar987]

That is a very interesing idea but I don't understand the main argument. Does the exstence of a nowhere vanishing vector field on S² implies the contractibility of the tangent circle bundle?

 

[lavinia]

That is a very interesing idea but I don't understand the main argument. Does the exstence of a nowhere vanishing vector field on S² implies the contractibility of the tangent circle bundle?

Not contractibility - but triviality. Since the sphere is orientable one non-zero vector field automatically gives two by rotating the first 90 degrees in the positive direction.

If the tangent circle bundle of the sphere is trivial, its first homology can not have any torsion.

 

[lavinia]

That is a very interesing idea but I don't understand the main argument. Does the exstence of a nowhere vanishing vector field on S² implies the contractibility of the tangent circle bundle?

It would also be interesting to see what other manifolds can be obtained from different attaching maps. Certainly the 3 sphere is one since it fibers over the 2 sphere by the Hopf fibration.

 

[lavinia]

Here's where I got to.

the coordinate transformation z ->1/z can be written in coordinates as

u = x/(x^2 + y^2) v = -y/(x^2 + y^2)

The Jacobian for points on the unit circle is

d/dx = (y^2 - x^2)d/du - 2xyd/dv

d/dy = 2xyd/du + (y^2 - x^2)d/dv

This is the a rotation through an angle whose sine is 2xy.
Note that if (x,y) rotates around the unit circle once, the image of d/dx circles twice around the unit circle in the (d/du,d/dv) plane.The tangent circle bundle of the 2 sphere falls into two solid tori that are glued together along their boundary tori above the unit circle. The Jacobian is the gluing map,

The two generators of the fundamental group of the bounding torus in the xy plane are
d/dx is held constant and (x,y,) is held constant. The first is null homotopic in the solid torus, the second is not. If their homology class are a and b respectively ( a is the null homotopic circle) and the corresponding classes in the other solid torus are c and d then the Jacobian maps a to c + 2d. So 2d is null homotopic which seems to says that the first homology of the tangent circle bundle is Z/2Z. Needs more detail.

 

[lavinia]

Here's where I got to. The Jacobian for points on the unit circle is

d/dx = (y^2 - x^2)d/du - 2xyd/dv

d/dy = 2xyd/du + (y^2 - x^2)d/dv

I think I may have a sign error here. I think the signs of 2xy need to be reversed.

I think a similar argument should work for the Hopf fibration. Here though one already knows that the circle bundle is S^3. still the solid tori gluing should show that the resulting manifold is simply connected. I wonder if a is identified with c + d.

 

[lavinia]

Here is another thought.

The sphere has a connection of constant postitive curvature,1.

The exterior derivative of the connection one form on the tangent circle bundle equals minus the pull back (under the bundle projection map ) of the volume form of the sphere.

Therefore, if there were a non zero vector field the exterior derivative of the connection one form would pull back to the volume element of the sphere. This is not possible because the volume element of the sphere is not exact.

By the same reasoning the curvature of every connection on the torus must have points where it is zero because the tangent bundle of the torus is trivial.

So it seems that the Gauss Bonnet theorem is overkill if you just want to prove that the sphere has no non-zero vector field. You do not need to know that the index of every vector field is the Euler characterisitic.

 

[lavinia]

Here's where I got to.

the coordinate transformation z ->1/z can be written in coordinates as

u = x/(x^2 + y^2) v = -y/(x^2 + y^2)

The Jacobian for points on the unit circle is

d/dx = (y^2 - x^2)d/du - 2xyd/dv

d/dy = 2xyd/du + (y^2 - x^2)d/dv

This is the a rotation through an angle whose sine is 2xy.
Note that if (x,y) rotates around the unit circle once, the image of d/dx circles twice around the unit circle in the (d/du,d/dv) plane.


The tangent circle bundle of the 2 sphere falls into two solid tori that are glued together along their boundary tori above the unit circle. The Jacobian is the gluing map,

The two generators of the fundamental group of the bounding torus in the xy plane are
d/dx is held constant and (x,y,) is held constant. The first is null homotopic in the solid torus, the second is not. If their homology class are a and b respectively ( a is the null homotopic circle) and the corresponding classes in the other solid torus are c and d then the Jacobian maps a to c + 2d. So 2d is null homotopic which seems to says that the first homology of the tangent circle bundle is Z/2Z. Needs more detail.

One can also see the attaching map in complex coordinates as d/dz -> (-1/z^2) d/dw

where w = 1/z are the coordinates around infinity. On the unit circle this is

d/dz -> -(e^-i2a)d/dw where z = e^ia so the double rotation is evident.

Equivalently, choose any tangent vector at the north pole of the sphere and parallel translate it along great circles to a small circle near the south pole.The vector field along the boundary has index 2 around the south pole

 

[lavinia]

Equivalently, choose any tangent vector at the north pole of the sphere and parallel translate it along great circles to a small circle near the south pole.The vector field along the boundary has index 2 around the south pole

Continuing the parallel translation all the way to the south pole sweeps out a projective plane whose Z boundary is twice the fiber circle over the south pole. This shows that the tangent circle bundle is not trivial for if it were then no multiple of any fiber would be a boundary.

 

[lavinia]

my favorite is lefschetz's proof. chose any point on the sphere where the vector is not zero. remove a small disk about that point. a bit of thinking will convince you that the degree of the map defined by the vector field on the boundary of the complementary disc is 2, hence on the complement of the disc around that point, there are 2 zeroes counting multiplicities. this takes a little imagination to visualize it but it is very elementary.

What if I take the vector field that tangent to geodesic flow from the north to the south pole. There are two singularities each with degree one. I think you mean the vector field obtained by parallel translating a fixed vector along great circles.

 

[mathwonk]

I do not understand. in your example there are indeed two zeroes counting multiplicities, i.e. you have two zeroes eachg with multipliciy one, and 1+1 = 2.

The argument is to look near any point where the vector is not zero, hence nearby the vectors are all parallel roughly to that one. assume all these vectors are roughly horizontal, and point say to the right. Now the complement of a small disc about that point is also a disc and has a non zero vector at each boundary point.

Then the boundary circle, viewed as the boundary circle of the complementary disc, has both vectors at top and bottom of the disc still parallel. but on the sides, the vector on the left points out of the complementary disc, while on the right side the vector points into the complementary disc. If you think about it, this means the vector field on the boundary circle has revolved 360 degrees on each half circle, hence has winding number 2 on the complementary boundary circle. that implies the sum of the degrees at the zeroes of the vector field on the complementary disc is 2, if there is only a finite number of zeros.

sorry if this is not clear, but it has nothing to do with parallel translation as far as I can see.

 

[zhentil]

It would also be interesting to see what other manifolds can be obtained from different attaching maps. Certainly the 3 sphere is one since it fibers over the 2 sphere by the Hopf fibration.

You can get Z of them, I believe :) Look at the degree of the map from the equator to U(1).

 

[lavinia]

I do not understand. in your example there are indeed two zeroes counting multiplicities, i.e. you have two zeroes eachg with multipliciy one, and 1+1 = 2.

The argument is to look near any point where the vector is not zero, hence nearby the vectors are all parallel roughly to that one. assume all these vectors are roughly horizontal, and point say to the right. Now the complement of a small disc about that point is also a disc and has a non zero vector at each boundary point.

Then the boundary circle, viewed as the boundary circle of the complementary disc, has both vectors at top and bottom of the disc still parallel. but on the sides, the vector on the left points out of the complementary disc, while on the right side the vector points into the complementary disc. If you think about it, this means the vector field on the boundary circle has revolved 360 degrees on each half circle, hence has winding number 2 on the complementary boundary circle. that implies the sum of the degrees at the zeroes of the vector field on the complementary disc is 2, if there is only a finite number of zeros.

sorry if this is not clear, but it has nothing to do with parallel translation as far as I can see.

I guess I didn't understand. Sorry. I just thought that each zero only winds once on each 2 disc not twice. It will wind twice on the only complementary 2 disc if there is only one zero.
Parallel translation of a single vector would create a single zero with multiplicity 2 and defines a null homotopy of the doubly winding curve.

 

[mathwonk]

yes, that's exactly it, the idea is to look at the complement of a disc on which there are NO zeroes. you might enjoy reading lefschetz's better description than mine.

you have to have an argument that works without assuming any zeroes exist.

 

[lavinia]

yes, that's exactly it, the idea is to look at the complement of a disc on which there are NO zeroes. you might enjoy reading lefschetz's better description than mine.

you have to have an argument that works without assuming any zeroes exist.

On the other hand, on the boundary of an open ball around a zero, the vector field winds around the surface of a solid torus. It one thinks of the fiber at the zero as the central meridian of the solid torus, then the index of the vector field at the zero is the winding number of the vector field around the meridian.

 

[lavinia]

I do not understand. in your example there are indeed two zeroes counting multiplicities, i.e. you have two zeroes eachg with multipliciy one, and 1+1 = 2.

In my example there is only one zero with multiplicity 2.

The argument is to look near any point where the vector is not zero, hence nearby the vectors are all parallel roughly to that one. assume all these vectors are roughly horizontal, and point say to the right. Now the complement of a small disc about that point is also a disc and has a non zero vector at each boundary point.

ok

Then the boundary circle, viewed as the boundary circle of the complementary disc, has both vectors at top and bottom of the disc still parallel. but on the sides, the vector on the left points out of the complementary disc, while on the right side the vector points into the complementary disc. If you think about it, this means the vector field on the boundary circle has revolved 360 degrees on each half circle, hence has winding number 2 on the complementary boundary circle. that implies the sum of the degrees at the zeroes of the vector field on the complementary disc is 2, if there is only a finite number of zeros.

This is confusing. Can you explain it again?

... it has nothing to do with parallel translation as far as I can see.

agreed

 

[lavinia]

As this thread unfolds I am learning more that I originally hoped for about the 2 sphere.

I just read a proof in Milner's Topology from the Differentiable Viewpoint that the sum of the indices of a vector field on a manifold is the same for all vector fields.

He embeds the manifold in Euclidean space and the extends the vector field to a vector field on a tubular neighborhood of the embedded manifold that points outward along the boundary and has the same zeros with the same indexes as the original vector field. The construction is straight forward.

By cutting out small balls around the zeros then normalizing the vector field to have unit length, he gets a map of the tube minus the open balls into the unit sphere.

The degree of this map on the boundary must be zero so the sum of the indices of the vector field equals the degree of the map of the boundary of the tube into the sphere.

For a hypersurface of Euclidean space, the tube has two components and the map into the sphere on each component has the same degree. This means that the Euler charateristic of a manifold that can be embedded as a hypersurface of Euclidean space is even. So every orientable 2 dimensional surface has even Euler characteristic. For the 2 sphere these maps have degree 2.

 

[mathwonk]

When I said you had two zeroes, each of degree one, I was quoting your own statement in #17.

Cut out a small disc from the sphere, centered at a point where there is a non zero tangent vector, and on which all vectors are roughly horizontal, pointing to the right. Now look at the complement, a large sphere with a small missing disc.In order to view that large punctured sphere as a disc, think of it as rubber, and let the rubber contract. The large outside part of the sphere snaps into place inside the small boundary circle. As it does this, some of the vectors on the boundary circle, in order to remain tangent to the surface flip over.

The vectors at top and bottom stay fixed pointing right, but the vectors at 3 o'clock and 9 o'clock flip 180 degrees and now point left. The other vectors flip differently according to the angle they make with the boundary circle. I.e. we are "reflecting" the part of the sphere outside the boundary circle to the inside. So vectors tangent to the boundary circle are left fixed, and the others are reflected in the tangent line to the boundary circle.

Thus the more perpendicular they are to the boundary circle the more they change direction. Since the angle between the previously horizontal vectors and the tangent lines to the circle, go through two full cycles as you go around the circle, the winding number, i.e. the total degree, of the vector field on the boundary circle of the new disc, is two. Hence either there is an infinite number of zeroes inside the new disc, or there are a finite number with the sum of the degrees equaling 2.

Either way there is a zero inside the new disc, i.e. outside the old disc.

 

[mathwonk]

By the way this is also a proof that the total sum of the degrees of a vector field on the sphere with a finite set of zeroes, is 2. In general, as you read in Milnor, the sum is the Euler characteristic, which is called the Poincare - Hopf theorem. Equivalently, on a Riemann surface, Riemann already proved the sum of the degrees of the zeroes of a non trivial analytic covector field is 2g-2, or minus the euler characterstic, since it is a co-vector field, or a section of the dual bundle to the tangent bundle.

 

[mathwonk]

OK now I see what this has to do with parallel translation by reading example 11.18 p. 125 of Bott -Tu. If you have access to this book, notice there they are constructing a smooth vector field not on the whole sphere, but only on the complement of the south pole. Note however that the picture there in fig. 11.2, actually is exactly what I tried to describe as the flipped vector field on the boundary circle of the complementary disc in my proof.

what puzzled me was how you were going to get a smooth vector field on the whole sphere this way, since parallel translation preserves length, so there would be no zeroes at all. but looking at that picture you can probably imagine how to extend that vector field to one on the excised disc, by shrinking the lengths of the translated vectors to zero as you approach the south pole. it may not be smooth at the south pole but it will at least be continuous, which should be enough.

However they are seeking a section no the whole sphere, just a non zero section on the complement of the south pole. So actually this is essentially the same proof I described, except you have to believe their non zero section extends to the south pole, picking up a zero, of degree 2 as you said.

I.e. what they call a "section" is apparently a never zero section. So, in theorem 11.16, they are doing exactly the same thing Lefschetz did, namely calculating the degree of the zeroes by looking at the field on the complement of the zeroes. You still have to be able to visualize it to believe their picture in fig. 11.2 is correct, i.e. gives degree 2. The connection is that by parallel translating, they get essentially the parallel vector field I started with on my original small disc. I.e. my small disc is their large complement of a disc around the south pole.

 

[mathwonk]

Milnor's book is superb for insight, my favorite. It cannot be beat for content per page. For more technical details on things like the Hopf theorem, Guillemin and Pollack expand on the excellent but brief presentation of Milnor. Guillemin and Pollack use over three times the number of pages, and sometimes some of Milnor's insight is lost, (they make the fundamental theorem of algebra look harder than necessary) but it is a nice complement to Milnor, and they make the technicalities look irresistibly easy.

 

[lavinia]

When I said you had two zeroes, each of degree one, I was quoting your own statement in #17.

Cut out a small disc from the sphere, centered at a point where there is a non zero tangent vector, and on which all vectors are roughly horizontal, pointing to the right. Now look at the complement, a large sphere with a small missing disc.


In order to view that large punctured sphere as a disc, think of it as rubber, and let the rubber contract. The large outside part of the sphere snaps into place inside the small boundary circle. As it does this, some of the vectors on the boundary circle, in order to remain tangent to the surface flip over.

The vectors at top and bottom stay fixed pointing right, but the vectors at 3 o'clock and 9 o'clock flip 180 degrees and now point left. The other vectors flip differently according to the angle they make with the boundary circle. I.e. we are "reflecting" the part of the sphere outside the boundary circle to the inside. So vectors tangent to the boundary circle are left fixed, and the others are reflected in the tangent line to the boundary circle.

Thus the more perpendicular they are to the boundary circle the more they change direction. Since the angle between the previously horizontal vectors and the tangent lines to the circle, go through two full cycles as you go around the circle, the winding number, i.e. the total degree, of the vector field on the boundary circle of the new disc, is two. Hence either there is an infinite number of zeroes inside the new disc, or there are a finite number with the sum of the degrees equaling 2.

Either way there is a zero inside the new disc, i.e. outside the old disc.

This proof I think is the same as the one I gave with parallel translation of a single vector across the sphere.

I think that another version of the same thing is to calculate the index of a constant vector field on the plane at infinity.

 

[mathwonk]

have you read posts 27,28? do you see what i mean about parallel translation not giving a vector field on the sphere?

maybe i misunderstood you as your earlier post does say the vector field goes up to a circle around the south pole. but then you have to prove that this partial vector field suffices to prove something. perhaps you have done this previously. anyway, if what you meant was that the horizontal vector field I described inside the small disc is equivalent to one obtained by parallel translating a vector from a pole, that is true. Still my argument starts from a given vector field and proves it has a zero. Yours constructs a partial vector field with no zero. The conclusion of your argument hence is that any extension of your vector field to the whole sphere must have a zero, but i do not see how that proves the result.

I.e. you still need to prove such an extension exists, plus you must prove that all vector fields have the same total degree. It seems to me this is far more than needed for the result, as Lefschetz's simple argument shows. I like seeing that they are in essence the same though.

 

[mathwonk]

lavinia maybe you are taking for granted all the background in a book like Bott -Tu. If so all these extra matters I mention may seem trivial to you. E.g. on p. 71, they show how to use a "bump" function to make a smooth extension of a radius function. Then in their proof on page 124 of the degree result, they appeal to this construction of a "global angle form". That is why it takes them 129 pages to give the proof. I am just saying none of this machinery is needed for Lefschetz's argument, beyond the idea that if there are no zeroes inside the circle then the degree on the circle itself would be zero. But in his computation it is visibly 2, for any vector field with finitely many zeroes. done.

Everything you are doing looks great and seems correct. I am just pointing out how simple Lefschetz makes this argument look. In his formulation it can be given to almost anyone. (Even I can understand it and i don't even know what a Thom class is.)

 

[lavinia]

lavinia maybe you are taking for granted all the background in a book like Bott -Tu. If so all these extra matters I mention may seem trivial to you. E.g. on p. 71, they show how to use a "bump" function to make a smooth extension of a radius function. Then in their proof on page 124 of the degree result, they appeal to this construction of a "global angle form". That is why it takes them 129 pages to give the proof. I am just saying none of this machinery is needed for Lefschetz's argument, beyond the idea that if there are no zeroes inside the circle then the degree on the circle itself would be zero. But in his computation it is visibly 2, for any vector field with finitely many zeroes. done.

Everything you are doing looks great and seems correct. I am just pointing out how simple Lefschetz makes this argument look. In his formulation it can be given to almost anyone. (Even I can understand it and i don't even know what a Thom class is.)

I just redid the Lefschetz proof for myself in the gym and see its elegance. Take a vector field and rotate the sphere so that the field at the north pole is not zero. Map the field minus a small polar ice cap into the plane by stereographic projection. The zeros are trapped inside a large disc and on the boundary you have the image of an approximately constant vector field. If you normalize the image field to have length one you get a map of the disc into the unit circle. The usual Milner/Hopf argument which I mentioned above now applies and tells you that the sum of the indices of the zeros inside the disc equals the index of a constant vector field at infinity - which is 2. Very cool.

 

[mathwonk]

sounds like you get it more clearly than i do. great! i loved it when i read it sitting in the library, some 40 years ago. It took me a while to visualize it though. Lefschetz was a neat guy, very upbeat, in spite of having lost his hands in an accident as an engineer, nothing to do but become a great pure mathematician. He looked kind of like a cheerful Dr. No with prosthetic hands. There is cool little AMA (or AMS?) movie of him explaining his fixed point theorem.

 

[mathwonk]

of course this proof puts all the heat on the basic theorem that a smooth map of a disc to the circle must have winding number zero on the boundary circle. I had fun making up a proof of this as a young teacher, using stokes theorem, which i now know is the standard proof. at the time i was just trying to think of some way to apply stokes theorem, for my class. i wound up presenting brouwer fixed point, fundamental theorem of algebra, and the vector field theorem, to my advanced calculus class. this sort of thing is standard now too.

in the plane all you have to know is stokes theorem, plus how to integrate dtheta around a circle. (Bott - Tu's global angle form). In three space you use the analogous solid angle form, e.g. in spherical coordinates. I never understood why books like thomas' calculus presented calculations with all these technical integrals and never used them for anything interesting.

One thing kind of interesting here is that in the plane this is a mod 2 problem but in the three space it is subtler. I.e. the fixed point theorem only requires the fact that 1 ≠ 0, while the degree theorem requires that 1 ≠ -1, much more difficult. I.e. you can do the first with mod 2 homology or intersection numbers, but not the second.

 

[mathwonk]

by the way thanks very much for the discussion. i now understand how lefschetz's hands on proof fits into these other ideas MUCH more than I did before.