Re: Partitions of unity proof roadblock

On Dec 2, 3:27 am, smn <smnewber...@xxxxxxxxxxx> wrote:
On Dec 1, 11:29 pm, smn <smnewber...@xxxxxxxxxxx> wrote:



On Dec 1, 4:40 pm, Jeff Rubin <JeffBRu...@xxxxxxxxx> wrote:

On Dec 1, 2:36 pm, smn <smnewber...@xxxxxxxxxxx> wrote:

On Dec 1, 8:28 am, Jeff Rubin <JeffBRu...@xxxxxxxxx> wrote:

On Nov 30, 5:21 pm, smn <smnewber...@xxxxxxxxxxx> wrote:

On Nov 30, 4:27 pm, Jeff Rubin <JeffBRu...@xxxxxxxxx> wrote:

On Nov 30, 3:08 pm, smn <smnewber...@xxxxxxxxxxx> wrote:

On Nov 30, 10:59 am, Jeff Rubin <JeffBRu...@xxxxxxxxx> wrote:

I recently posted about the trouble I was having understanding
a proof in Lang's Fundamentals of Differential Geometry relating
to partitions of unity. So I tried turning to another text,
Manifolds, Tensor Analysis, and Applications by Abraham,
Marsden, and Ratiu. This text had a different approach but
my understanding ran into the same roadblock. I was wondering
if someone could help me out and explain.

The proof is of proposition 5.5.17 where M is a paracompact manifold
modeled on a Banach space E. It shows the equivalence of a number
of statements, the last two of which are:
(iv) every chart domain of M admits C^k partitions of unity
subordinate
to any open covering
(v) E admits C^k partitions of unity subordinate to any open covering
of E.

I had no problem with the implications (i)=>(ii)=>(iii)=>(iv). But
the proof
of (iv)=>(v) has got me stumped:

I wanted to describe a simplified version of the proof which pointed
out the problem I'm having, but I can't figure out how to do that
without sounding very confused. So please bear with me; here
is the full proof directly from the text: (I use /\ for intersection
and \/ for union)

"Consider now any open covering {U_\alpha}_{\alpha \in A} of E and let
(U, \phi) be an arbitrary chart of M. Refine first the covering of E
by taking
the intersections of all its elements with all translates of \phi(U).
Since E
is paracompact, refine again to a locally finite open covering {V_
\beta}.

When you do this you can refine by open balls whose closure B in E
(the closed balls )are still
contained in U .The function g of the partition of unity below that
worries you is 0 on E\B (open in E ) and C^k on U (open in E) and
hence C^k in E

Regards,smn

The inverse images by translations and \phi of these open sets are
subsets of U,
hence chart domains, and thus by (iv) they admit partitions of unity
subordinate
to any open covering, for example to {V_\beta /\ U_\alpha : \alpha \in
A};
call it {g_i^\beta}. Then g = \sum_{i,\beta} g_i^\beta is a C^k map
and the
double-indexed set of functions g_i^\beta / g forms a C^k partitions
of unity
of E."

Now I follow this down to and including that {g_i^\beta} is a
partition of unity
subordinate to the mentioned covering of V_\beta. Now these functions
g_i^\beta are necessarily C^k maps from V_\beta into R. They are not
(yet) maps from E into R. To get that you would have to extend them
by assigning the value 0 to all the points of E - V_\beta. You then
have
to show that the resulting extension is C^k. This is fine on points
in V_\beta
and on points outside of the closure of V_\beta, but gives me trouble
on
the boundary of V_\beta. There is no guarantee that the locally
finite covering
of V_\beta (that comes with the partition of unity {g_i^\beta}) is
locally finite
with respect to any points outside of V_\beta.

How do you fix this?

Thanks,

Jeff

Thanks for responding. I know what you are getting at, but the
details
still leave me puzzled. I am familiar with the result that in a
normal
space you can shrink a locally finite open covering of a set to one
whose
closures are contained in the corresponding open sets of the original
covering.
In the proof given in the cited book, it is not clear where doing this
shrinking helps. Once you have a manifold domain and a covering, you
get a partition of unity subordinate to that covering;
that is, a locally finite open refinement and the C^k functions which
sum to one and whose supports
lie in the locally finite open covering. If you try to shrink the
covering
that is part of the partition of unity, then what do you do to the
functions
to regain the support requirement? I'm still so confused.

Jeff- Hide quoted text -

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The point is to get the closures IN E (not just in U) to be in the
cordinate domain U .I think you can do that in the original
construction in U ,using balls as I said .No shrinking theorem used
later .Regards smn

Okay. I can see how to get the g_i^\beta to extend to
all of E and still be C^k. Thanks for that bit of the
puzzle. But I am immediately stymied by the next step,
which is to form the double sum over i and \beta of
g_i^\beta and to show that is C^k. For each point in E,
I need to find a neighborhood of the point which
intersects at most finitely many of the carriers of
g_i^\beta (to show that the sum is finite). So suppose
you have a point on the boundary of some V_\beta.
What guarantees that it has a neighborhood that
intersects at most finitely many of the carriers of
g_i^\beta? It cannot be the local finiteness of the
partition of unity transported from the manifold
chart domain to V_\beta since that is only locally
finite with respect to the open set V_\beta. I had
the same problem with Lang's proof that I referenced
in my other post (Incorrect proof in Lang's Fundamentals
of Differential Geometry).

Help, again, please.

Thanks,

Jeff- Hide quoted text -

- Show quoted text -

I am not sure I see the problem.Your origional cover of E was refined
to be locally finite so for each point of E there should be a finite
number of tranlates of the open chart domains intersecting the point

Yes, I'm with you so far

and then for each such chart a ball intersecting only a finite number
of supports of the
functions

It's true that for a point of E which is in the translate
of a chart domain, it will have a neighborhood intersecting
only a finite number of supports of the functions. But there
is no such guarantee for any point of E which is on the boundary
of a translate of a chart domain.

Every point of E is in the translate of a chart domain .Boundaries are
not relevant -the functions involved in the sum are all c^k on ALL of
E .The small ball about the point contains points of the support of
only a finite number of the functions so the infinite sum when
restricted to the ball is a finite sum (almost all the functions are 0
everywhere on the ball )so the sum is a c^k function on the ball and
positive at the center .THis true for all the balls the infinite sum
defines an everywhere positive function .Deviding each function of the
sum by this positive function gives the required partition of 1 .smn

so on the intersection of these balls (open) only a finite
number of functions are non zero. So the sum of a finite number of C^k
functions is C^k .What am I missing ?

To prove that the sum is C^k, you have to find a neighborhood of
each point and show that the function is C^k on that neighborhood.
To do that, you have to first find a neighborhood of each point
where the definition of the function has only a finite number of
terms. I can't figure out how to find such a neighborhood for
a point on the boundary of a translate of a chart domain.

I thought of something else .Isn't an open ball in a Banach Space
diffeomorphic to the whole Banach space ? That would simplify the
whole problem.smn

It is true, but I'm pretty far from seeing how it helps right now.

Thanks,

Jeff- Hide quoted text -

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Hi ,I looked at the Marsden proof again.Yes I think now that it is
incomplete or incorrect.You have to choose the locally finite
refinement of the cover U_alpha more carefully then just intersections
with translates of phi(U)along the lines I stated earlier namely try
for each point x of e choose a ball whose closure in E (a closed ball)
is contained in the intersection of some U_alpha with some translate
of phi(U) .Now take a locally finite refinement of all these balls and
then get your gib 's from this refinement and then all your objections
should disappear by my previous arguments.
Hopefully,smn

I have no problem with the construction as you have stated it.
I wind up with the {V_\beta} being a locally finite open cover
of E with each V_\beta contained in some open ball whose
closure is contained in some U_\alpha and also in some
translate of \phi(U). The next step is to do the reverse
translation of V_\beta and then take its inverse image under
\phi. This gets me an open subset of the chart domain U.
By (iv) I can get a partition of unity for this subset
subordinate to any given open covering of the subset and
then translate it and "\phi" it to get a partition of unity
of V_\beta subordinate to a given open covering of V_\beta. No
matter what covering I use (call it {C_\delta} for now),
I am going to get a partition of unity {(G_i^\beta, g_i^\beta)}
subordinated to {C_\delta}, where {G_i^\beta} is locally finite
only with respect to V_\beta (emphasis on this last phrase).
Although these g_i^\beta are only defined on V_\beta, I agree
there is no problem extending them to be C^k on all of E.
However, if you take a point e on the boundary of V_\beta,
there is still no guarantee that it has any neighborhood
at all which intersects only a finite number
of G_i^\beta. So when I try to prove \sum g_i^\beta is C^k
at e, I can't find a neighborhood of e where the sum is even
defined. It doesn't matter how small I make V_\beta, since
the G_i^\beta are a locally finite covering of V_\beta
and the local finiteness is only with respect to V_\beta.
I'm still stuck in the same way as before.

Jeff
.

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