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Topology on space of hyperfunctions

This is a reference request, coming from someone with little knowledge of hyperfunctions: Which methods have been used to endow the space of hyperfunctions $\mathcal B(\mathbb R)$ with something like ...
Peter Scholze's user avatar
7 votes
2 answers
359 views

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 ...
Brian Lins's user avatar
3 votes
1 answer
133 views

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-...
cs89's user avatar
  • 981
3 votes
1 answer
124 views

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 ...
Paul's user avatar
  • 1,379
4 votes
0 answers
95 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
  • 2,591
6 votes
0 answers
155 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
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
2 votes
1 answer
256 views

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 ...
Thomas Kurbach's user avatar
5 votes
1 answer
221 views

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 ...
Keivan Karai's user avatar
  • 6,162
1 vote
0 answers
74 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
  • 4,113
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 $$\...
Ali Taghavi's user avatar
2 votes
1 answer
175 views

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 $I$ ...
JBrude's user avatar
  • 115
3 votes
0 answers
294 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 ...
user151260's user avatar
3 votes
1 answer
200 views

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{...
Guilia S's user avatar
  • 105
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(...
S. Dewar's user avatar
  • 266
7 votes
2 answers
438 views

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. ...
Arrow's user avatar
  • 10.4k
1 vote
1 answer
134 views

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 $f:U_\epsilon(0)\to\mathbb{R}$ ...
ar.grig's user avatar
  • 1,133
1 vote
0 answers
63 views

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 $C^\infty$-...
ar.grig's user avatar
  • 1,133
3 votes
1 answer
164 views

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 $x_{n_k}$ ?...
ar.grig's user avatar
  • 1,133
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
  • 1,379
2 votes
1 answer
131 views

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 ...
Ali Taghavi's user avatar
4 votes
2 answers
486 views

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 $\...
cll's user avatar
  • 2,305
11 votes
2 answers
924 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 exist ...
Asaf Shachar's user avatar
  • 6,641
6 votes
0 answers
119 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
  • 21.3k
4 votes
2 answers
193 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, ...
Ali Taghavi's user avatar
2 votes
0 answers
99 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
5 votes
2 answers
362 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 ...
Anonymous Coward's user avatar
9 votes
0 answers
957 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 ...
Omar's user avatar
  • 435
3 votes
1 answer
427 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 ...
Alec Payne's user avatar
20 votes
1 answer
2k 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 ...
Grisha Papayanov's user avatar
6 votes
0 answers
233 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 ...
user avatar
14 votes
1 answer
826 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 ...
Joonas Ilmavirta's user avatar
1 vote
0 answers
200 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-...
Feanoris's user avatar
2 votes
1 answer
127 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 ...
Olorin's user avatar
  • 501
1 vote
0 answers
172 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
6 votes
1 answer
302 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 ...
Dima Sustretov's user avatar
3 votes
0 answers
340 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
6 votes
1 answer
508 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) \...
peter's user avatar
  • 211
6 votes
0 answers
265 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 ...
jorst's user avatar
  • 359
6 votes
0 answers
268 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
  • 1,020
15 votes
3 answers
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 ...
H. Berbeleque's user avatar
1 vote
2 answers
144 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 ...
Ali Taghavi's user avatar
6 votes
2 answers
3k 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 ...
jorst's user avatar
  • 359
20 votes
2 answers
2k 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 ...
Michael Albanese's user avatar
1 vote
0 answers
160 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
  • 85
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
16 votes
3 answers
1k 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 ...
Igor Khavkine's user avatar
4 votes
2 answers
478 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 ...
Ali Taghavi's user avatar
2 votes
0 answers
303 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