Questions tagged [real-analytic-structures]

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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 ...
Omar's user avatar
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6 votes
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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 ...
Eduardo Longa's user avatar
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 $A$. ...
asv's user avatar
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6 votes
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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 ...
user avatar
6 votes
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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 ...
jorst's user avatar
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6 votes
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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 ...
Mattia Talpo's user avatar
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4 votes
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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 ...
Rami's user avatar
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4 votes
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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 $$\...
Ali Taghavi's user avatar
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 $p$ ...
Paul's user avatar
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3 votes
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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 ...
user151260's user avatar
3 votes
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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(...
S. Dewar's user avatar
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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 ...
user284045's user avatar
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 $g$ ...
Ali Taghavi's user avatar
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) ...
Anonymous Coward's user avatar
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 $\...
Ali Taghavi's user avatar
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. ...
Eduardo Longa's user avatar
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 ...
William of Baskerville's user avatar
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 ...
Ali's user avatar
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1 vote
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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-...
Feanoris's user avatar
1 vote
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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 ...
peter's user avatar
  • 211
1 vote
0 answers
152 views

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.
tujunwu's user avatar
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