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From: Rupert on 9 Aug 2010 22:32
On Aug 10, 7:19 am, bacle <h...(a)here.com> wrote:
> > On Aug 9, 1:34 pm, bacle <h...(a)here.com> wrote:
> > > Hi:
> > >  Given a complex n-manifold M, i.e., complex
> > dimension n,
>
> > >  is there a "natural", or "canonical" way of giving
> > M a
>
> > >  real 2n-structure.?
>
> > What degree of "smoothness" do you want?
>
>    Thanks. I was hoping thatif M had a complex structure,
>    then M with the real structure would automatically be
>   C^oo . Isn't it.?
>

Yes, in fact it is even C^omega.

>        Also: could you please suggest how to go in the opposite
>     direction, i.e., given M a real, orientable, 2n-dimensional
>     manifold: how would we give M a complex structure.?

Do you understand what is meant by a C^omega differentiable structure?
From: Rupert on 10 Aug 2010 02:53
On Aug 10, 1:49 pm, bacle <h...(a)here.com> wrote:
> > On Aug 10, 7:19 am, bacle <h...(a)here.com> wrote:
> > > > On Aug 9, 1:34 pm, bacle <h...(a)here.com> wrote:
> > > > > Hi:
> > > > >  Given a complex n-manifold M, i.e., complex
> > > > dimension n,
>
> > > > >  is there a "natural", or "canonical" way of
> > giving
> > > > M a
>
> > > > >  real 2n-structure.?
>
> > > > What degree of "smoothness" do you want?
>
> > >    Thanks. I was hoping thatif M had a complex
> > structure,
> > >    then M with the real structure would
> > automatically be
> > >   C^oo . Isn't it.?
>
> > Yes, in fact it is even C^omega.
>
> > >        Also: could you please suggest how to go in
> > the opposite
> > >     direction, i.e., given M a real, orientable,
> > 2n-dimensional
> > >     manifold: how would we give M a complex
> > structure.?
>
> > Do you understand what is meant by a C^omega
> > differentiable structure?.
>
>   I think so; the coordinate-change functions are
>   real-analytic, isn't it.?.
>
>     I have been looking up results about extending
>   real-analytic functions f on intervals (-r,r)
>   ( work on (-r,r) for the sake of simplicity here)
>    then f can be extended (or, more precisely,
>   continued analytically) , into some domain
>   containing the disk |z|=r , but I have not been
>   able to generalize from this.
>
>   Thanks.- Hide quoted text -
>
> - Show quoted text -

If you know what it is for a real-valued function to be real-analytic
on an open subset U of a manifold which is homeomorphic to an open
subset V of R^2n then it shouldn't be too hard to define what it is
for a complex-valued function to be complex-analytic on U, viewed as
homeomorphic to an open subset W of C^n.
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