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The tubular neighbourhood theorem, stating that an embedded submanifold has a neighbourhood that is a diffeomorphic image of an open subset of the normal bundle, is a staple result about smooth ...
Peter McNamara's user avatar
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Let $(M,m_0)$ be a real analytic manifold. Let $\pi$ be a cotangent at $m_0$. How can one prove that there exists a global analytic function $\theta:M\rightarrow \mathbb{R}$ such that $d\theta(m_0)=\...
Amr's user avatar
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In, the paper Geometric categories and o-minimal structures by Van Den Dries and Miller, the definition of a Whitney stratification of a function is given for a function $f: A \rightarrow \mathbb{R}^n$...
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Say I have a complex analytic subspace $X$ of a complex manifold. Additionally: $X$ is a topological manifold, and For each $x \in X$, the set of derivatives at $x$ of smooth paths holomorphic discs ...
Alex Wright's user avatar
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I'm not sure this is a research-level question, but I couldn't find an answer after a bit of searching, so here goes. Let $f: \mathbb{R}^2 \to \mathbb{R}^2$ be a real-analytic function. Can we always ...
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Inspired by comment discussions in this MO post smooth version of splitting principle we ask: Are there two compact real analytic manifolds $M,N$ of dimension $m>n$ such that there is not any ...
Ali Taghavi's user avatar
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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
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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
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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
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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
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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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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 ...
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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
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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
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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 ...
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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
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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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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 $$\...
Ali Taghavi's user avatar
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I know that in general exact-by-exact extensions of $C^*$-algebras need not be exact. Is it true that, if we have a short exact sequence of $C^*$-algebras $$0 \to I \to A \to B \to 0$$ such that $I$ ...
JBrude's user avatar
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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 ...
Jimmy_the_analyst's user avatar
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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
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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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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
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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
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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
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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
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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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2 votes
1 answer
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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
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4 votes
2 answers
589 views

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
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12 votes
2 answers
1k views

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
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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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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, can ...
Ali Taghavi's user avatar
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2 votes
0 answers
121 views

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
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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
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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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1 answer
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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
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20 votes
1 answer
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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
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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
894 views

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
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1 vote
0 answers
272 views

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
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2 votes
1 answer
165 views

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
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1 vote
0 answers
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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
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6 votes
1 answer
430 views

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
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3 votes
0 answers
433 views

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
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6 votes
1 answer
568 views

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
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6 votes
0 answers
302 views

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
0 answers
330 views

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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16 votes
3 answers
1k views

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
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1 vote
2 answers
183 views

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
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6 votes
2 answers
4k views

(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
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