Questions tagged [real-analytic-structures]
The real-analytic-structures tag has no usage guidance.
21
questions with no upvoted or accepted answers
9
votes
0
answers
919
views
Complexification of smooth manifolds
Given a $C^\infty$- smooth manifold $M$, is there a natural way to complexify it?
By this I mean finding a complex manifold $N$ and a smooth function $f:M\to N$ such that $df:T_xM\otimes \mathbb{C}\to ...
6
votes
0
answers
149
views
Is the space of analytic sections of a vector bundle a Fréchet space?
Let $\pi : E \to M$ a smooth vector bundle of finite rank, where both $E$ and $M$ are finite dimensional smooth manifolds, and $M$ is compact. I already know that the space $\Gamma^\infty(M,E)$ of ...
6
votes
0
answers
118
views
Integrals of real analytic functions
Let $A\subset \mathbb{R}^{n+m}$ be a compact subanalytic subset. Let $F\colon A\to \mathbb{R}$ be a function which is a restriction to $A$ of a real analytic function defined in a neighborhood of $A$.
...
6
votes
0
answers
232
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Isomorphic copies of the real line--can these isomorphisms be made explicit?
This question has bugged me for a long time. I've asked some of my professors and they seem to believe that these objects haven't been studied. I'm prone to believe that the construction is too simple ...
6
votes
0
answers
262
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Does the cohomology comparison part of GAGA hold over the reals?
If $k=\mathbb{C}$, $X$ is a projective (or even proper) scheme over $k$ and $F$ a coherent sheaf on $X$, then there is an isomorphism
$$
H^p(X,F)\rightarrow H^p(X^{an}, F^{an})
$$
(and there is also ...
6
votes
0
answers
265
views
Where do I read about semi-algebraic/analytic sets?
What's a good introduction to semi-algebraic/semi-analytic sets?
I'm coming from algebraic geometry, so ideally I'd prefer material that has a similar point of view and highlights analogies.
I've ...
4
votes
0
answers
91
views
Holonomic distributions in the analytic setting
We are interested in a reference to the notion of Holonomic distributions in the real analytic setting. Namely, distributions that generate a Holonomic D-module under the action of the algebra of ...
4
votes
0
answers
148
views
The range of $\int_M \kappa_g ds$ where $g$ varies in all possible real analytic metrics on $M$
Let $M$ be a real analytic open surface(A non compact 2 dimensional manifold without boundary).
For every number $\lambda\in \mathbb{R}$, is there a real analytic Riemannian metric on $M$ with
$$\...
4
votes
0
answers
100
views
Flow lines of a real analytic vector field convergent to a point
Let $X$ be a real analytic vector field defined on an open connected subset $U$ of $\mathbb{R}^n$. Let $p \in \mathbb{R}^n \setminus U$. Let $L$ be union of the flow lines $\ell$ of $X$ such that $p$ ...
3
votes
0
answers
279
views
Real analytic function: zero set of the gradient is a subset of the zero set of the function
I had this question when reading Bierstone and Milman's famous paper "Semianalytic and subanalytic sets". In their proof of the Łojasiewicz gradient inequality (Proposition 6.8 in the paper), they ...
3
votes
0
answers
86
views
Is $|f^{-1}f(p)|$ constant on a conull set?
Suppose $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ is an analytic map. Can we say that there is a conull set $U \subset \mathbb{R}^n$ (i.e. $\mathbb{R}^n \setminus U$ has measure zero) where $|f^{-1}f(...
3
votes
0
answers
322
views
Lojasiewicz's structure theorem
The Lojasiewicz structure theorem on p. 169 in the book of Krantz/Parks A primer of real analytic functions confuses me. According to the stratification property, the zeroes of a real analytic ...
3
votes
0
answers
184
views
Characterization of certain analytic vector fields on $S^{2}$
Let $X$ be a real analytic vector field on $S^{2}$ which satisfies:
$X$ has a finite number of singularities on $S^{2}$
The equator is invariant under flow of $X$
3.$g_{*}X=\pm X$ where $g$ ...
2
votes
0
answers
98
views
Sheaves of functions on open semi-algebraic sets
Let $U\subsetneq\mathbb{R}^n$ be an open semi-algebraic set which is not Zariski open (the case I have in mind is that of an open ball $B(0,1)$). A function $f:U\rightarrow \mathbb{R}$ is called
(1) ...
2
votes
0
answers
299
views
Analytic version of the Cartan lemma
Assume that $\beta$ is a real analytic 2-form on an analytic manifold $M$ and $\alpha$ is an analytic non vanishing 1-form on $M$. Assume that $\beta \wedge \alpha=0$. Is there an analytic 1-form $\...
1
vote
0
answers
56
views
When does an analytic submanifold descend to the quotient?
Let $\pi : M \to B$ be a smooth principal bundle with group $G$, where $M$ is an analytic (Fréchet, in my case) manifold, $B$ is a smooth (Fréchet) manifold and $G$ is a smooth (Fréchet) Lie group. ...
1
vote
0
answers
65
views
Analytic maps via Baire category method
The Baire category method is sometimes useful in constructing smooth maps between manifolds with prescribed properties. I would like to know whether there are any (non-trivial) situations in which the ...
1
vote
0
answers
73
views
Smooth closed Riemannian manifolds with quasi-analytic metrics
I haven't found a reference for this exact definition in the literature so I wonder if what I want to pose here makes sense. I basically want to consider a smooth close Riemannian manifold that ...
1
vote
0
answers
195
views
Restriction of real analytic functions to embedded submanifolds
lets assume we have a real vectorspace $V$ and functions $f_1, \dots, f_k \colon V \to \mathbb{R}$ which are real-analytic (for instance, let them be polynomial).
Furthermore we have an embedded real-...
1
vote
0
answers
170
views
How many real-analytic forms exist on a real-analytic manifold?
I hope my question isn't too vague:
Let $M$ be a real-analytic compact manifold (say surface for simplicity). How big is $\Omega^k(M)$ (space of global real-analytic $k$-forms)?
Let me explain ...
1
vote
0
answers
152
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Does real analytic imply locally contractible?
The statement is true for complex analytic spaces. I am not sure who proved this result.
I ask the same question in the real case.