# Questions tagged [real-analytic-structures]

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

54
questions

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### Contractible real analytic varieties

If a real analytic variety $V$ in $\mathbb{R}^n$ is both bounded and contractible, is it true that $V$ must be a single point?
Here a real analytic variety is the set of zeros of a real analytic ...

3
votes

1
answer

128
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### Does Noetherianity imply division theorem?

I am trying to understand something which is probably basic for experts so I am sorry if this is not suited for this forum.
Let $\mathcal{O}_n$ denote the ring of germs at $0 \in \mathbb{R}^n$ of real-...

3
votes

1
answer

106
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### Analyticity of central stable manifolds

Let $X$ be a real analytic vector field defined on $\mathbb{R}^n$. Assume the origin $0 \in \mathbb{R}^n$ is a zero of $X$. Assume, furthermore, that we know that the center-stable manifold (in the ...

4
votes

0
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81
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### 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 ...

5
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0
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128
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### 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 ...

1
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0
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52
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### 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
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0
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65
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### 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 ...

2
votes

1
answer

222
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### Dimension of intersection of real analytic sets

Suppose that $A,B$ are real analytic subsets of $\Omega\subseteq \mathbb{R}^n$ and $p\in A\cap B \neq \emptyset$. Does the intersection inequality from complex analysis still hold, i.e. does the ...

5
votes

1
answer

211
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### When do volumes depend real-analytically on the parameters defining the regions?

Suppose $f_1, \dots, f_n$ are real-valued real analytic functions defined on an open set $B=(0,1)^d$ of $\mathbb{R}^d$.
For $r \in \mathbb{R}$, let $S_r$ be the sub-level set in $B$ defined by the ...

1
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0
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71
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### 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 ...

4
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0
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148
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### 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
$$\...

2
votes

1
answer

150
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### About nuclear-by-exact extensions

I know that in general exact-by-exact extensions of $C^*$-algebras need not be exact. Is it true that, if we have a short exact sequence of $C^*$-algebras
$$0 \to I \to A \to B \to 0$$
such that $I$ ...

3
votes

0
answers

255
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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 ...

3
votes

1
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167
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### Decomposition of a real analytic variety

Is the following true? If so, I would be grateful for a reference that contains such a result and its proof.
Let $f:\mathbb{R}^d\rightarrow \mathbb{R}$ be a real analytic function, and $V:=\{\mathbf{...

3
votes

0
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86
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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(...

6
votes

2
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383
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### In the real analytic category, are the fibers of a proper submersion isomorphic?

Ehresmann's theorem says that a proper smooth submersion is a fiber bundle. The proofs I know rely on the existence of connections locally on the base, and this is furnished by partitions of unity.
...

1
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1
answer

119
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### On a case of real-analytic interpolation

Given strictly increasing sequence $x_n$ of rational numbers with $\sup x_n = x$.
In which case (sufficient condition on $x_n$) there exists real-analytic function $f:U_\epsilon(0)\to\mathbb{R}$ ...

1
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0
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63
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### Real-analytic function with given set of values [closed]

We say that a strictly increasing sequence $x_n$ of reals converges fast
to $x$, if for each $k\in\mathbb{N}$ the sequence $n^k\cdot(x_n − x)$ is
bounded. It is known that there exists a $C^\infty$-...

4
votes

1
answer

154
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### Real analytic function of one variable with given set of values

Given two strictly increasing bounded sequences of reals $x_n$ and $y_n$. What is known about existence of real analytic function $f$ with property $f(x_{n_k})=y_{n_k}$ for some subsequence $x_{n_k}$ ?...

4
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96
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### 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 $p$ ...

2
votes

1
answer

129
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### A special oscillatory orbit in space

Edit: According to the comment of Prof. Eremenko I revise the question.
19 years ago, I have heard the following problem from a specialist of dynamical system. During these 19 years, I was in contact ...

4
votes

2
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429
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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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2
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832
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### 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 exist ...

6
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0
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115
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### 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 $A$.
...

4
votes

2
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186
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### 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, ...

2
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0
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97
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### 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) ...

5
votes

1
answer

303
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### 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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0
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857
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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 ...

3
votes

1
answer

399
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### 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 ...

19
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1
answer

1k
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### 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 ...

6
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0
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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 ...

14
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1
answer

784
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### 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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0
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181
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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-...

2
votes

1
answer

125
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### 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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0
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160
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### 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 ...

6
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1
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### 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 ...

3
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0
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289
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### 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 ...

6
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1
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478
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### 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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0
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254
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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
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### 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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3
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### 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 ...

1
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2
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143
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### 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 ...

5
votes

2
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3k
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### 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 ...

19
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2
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### 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 ...

1
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0
answers

142
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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.

3
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0
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184
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### 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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3
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1k
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### 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 ...

4
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2
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424
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### 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 ...

2
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0
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292
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### 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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2
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356
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### 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^{*}(...