# Questions tagged [real-analytic-structures]

The real-analytic-structures tag has no usage guidance.

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### Symplectic form on a Kähler manifold can be not real analytic?

Let $M$ be a Kähler manifold. The complex structure on it naturally gives rise to the real analytic structure. I wonder if there exist Kähler manifolds such that the associated symplectic $2$-form $\...

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482 views

### Are conformal maps between Riemannian manifolds real-analytic?

This is a cross-post.
Let $M,N$ be oriented smooth ($C^{\infty}$) $n$-dimensional Riemannian manifolds, and let $f:M \to N$ be a smooth orientation-preserving weakly* conformal map.
Do there ...

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89 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 $A$.
...

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140 views

### Polynomial vector field tangent to a given analytic simple closed curve

Assume that $\gamma$ is an analytic simple closed curve in $\mathbb{R}^2$ which surrounds origin.
Is there a polynomial vector field on the plane which is tangent to $\gamma$? In the other word, ...

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69 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) ...

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**1**answer

219 views

### Are continuous rational functions arc-analytic?

Let $X\subseteq\mathbb{R}^n$ be a smooth semi-algebraic set (for simplicity we can assume $X=B(0,r)$ is a small ball around the origin). A function $f:X\rightarrow \mathbb{R}$ is called a continuous ...

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82 views

### characterisation of definable functions in the Lipschitz-Robinson expansion

A subset $X$ of a domain $\Omega$ in a real analytic manifold $M$ is called semi-analytic if in a neighbourhood of any point $p \in M$ it is described by finitely many equations and inequalities in ...

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363 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}\...

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**1**answer

173 views

### Identity Theorem for Real-Analytic Hypersurfaces

There's an interesting statement it seems I can prove, but I can't find any references for it, which makes me suspicious of it. So, could someone verify that the statement is correct/incorrect or ...

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980 views

### When do real analytic functions form a coherent sheaf?

It is known that, in general, the sheaf of real analytic functions on a real analytic manifold is not coherent. However, there are some examples, where we have coherence: for example, if $X$ is a ...

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224 views

### 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 ...

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571 views

### Can a Riemannian metric be analytic in non-analytically different coordinates?

Suppose I have two coordinates on the same (subset of a) Riemannian manifold.
If the metric tensor is analytic in both coordinates, is the change of variables between them necessarily analytic?
In ...

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113 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-...

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**1**answer

116 views

### Polynomials pulled back by momentum maps

Let $G$ be a Lie group acting Hamiltonian on some real analytic symplectic manifold $(M, \omega)$, with an $G$-equivariant momentum map $\Phi \colon M \to \mathfrak{g}^*$.
Assuming I can find ...

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107 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 ...

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163 views

### Are definable sets in an o-minimal expansion of the real field locally analytic?

I have a strong suspicion that yes, but as I am not a specialist in o-minimal structures, I thought that I might have overlooked some corner case.
The precise statement is as follows: let $X \subset ...

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120 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 ...

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320 views

### Complexifying a real-analytic singularity

This is probably a well-known issue, but I could not find a clear discussion in the literature, and I think others could find it useful.
Consider a real-analytic function germ $f:(\mathbb R^2,0) \...

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213 views

### 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 ...

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184 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 ...

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1k views

### Can integration spoil real-analyticity?

Is there an example of a function $f:(a,b)\times(c,d)\to\mathbb{R}$, which is real analytic in its domain, integrable in the second variable, and such that the function
$$ g:(a,b)\to\mathbb{R},\qquad ...

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137 views

### Does an analytic tensorial Lie structure on $S^2$ gives a fiberwise Abelian Lie algebra structure?

Motivated by the answer to this question we ask:
Is it true to say that for every real analytic tensorial Lie algebra structure $\alpha$ on $\chi^{\infty}(S^2)$, all fibers are necessarily ...

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1k views

### Three and a half basic questions on the Weil restriction of scalars

(This is reposted from mathstackexchange, where it received no answer so far.)
I am currently trying to get familiar with the Weil Restriction functor.
For a finite field extension $L|K$ it ...

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**1**answer

678 views

### Is the analytic version of the Whitney Approximation Theorem true?

I initially asked this question on MSE but I haven't had any luck.
The Whitney Approximation Theorem states that any continuous map between smooth manifolds is homotopic to a smooth map. If the ...

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85 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.

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178 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 $g$ ...

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559 views

### Analog of Newlander–Nirenberg theorem for real analytic manifolds

It is well known that one can specify a complex structure on a real $C^\infty$ manifold in two equivalent ways: an atlas with holomorphic transition functions between charts and an integrable almost ...

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284 views

### analytic vector bundles

Let $E$ be a real analytic vector bundle on an analytic manifold $M$. Assume that $E$, as a smooth vector bundle, is a trivial bundle.
Is $E$ a trivial analytic vector bundle?
I need to the ...

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192 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 $\...

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270 views

### A cohomology associated to a 1- form

In this question all objects are real analytic.(manifolds, differential forms..)
Assume that $M$ is a compact manifold and $\alpha \in \Omega^{1}(M)$ is a one form.
We define a map $\phi:\Omega^{*}(...

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**1**answer

272 views

### Analytic vector fields on surfaces which have infinite number of singularities

Let $X$ be an analytic vector field on a compact oriantable surface $S$ with volume form $\omega$. We denote the set of its singularities by $Z(X)$.
A local question
Is there an analytic vector ...

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788 views

### Derivation on real analytic manifolds

Let $M$ be a real analytic manifold. By $C^{\omega}(M)$ we mean the algebra of all analytic functions from $M$ to $\mathbb{R}$. Assume that $D$ is a derivation on $C^{\omega}(M)$ .
Is there a ...

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585 views

### Number of disjoint simple closed geodesics

According to Jairo comment on the first version of this question I revise the question as follows;
Let $g$ be a real analytic Riemannan metric on $S^{2}$. Is it true to say that:
There are at most a ...

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796 views

### Linearization of vector fields

Simply, we discuss things on $C^{2n}$ ($or R^{2n}$) but not on manifolds. Given an anayltic (or real analytic) vector field $V$ on $C^{2n}$ ($or R^{2n}$), with a zero at the origin . And $V_{0}$ is ...