Questions tagged [real-analytic-structures]
The real-analytic-structures tag has no usage guidance.
54
questions
6
votes
2
answers
253
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 ...
- 540
3
votes
1
answer
128
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-...
- 948
3
votes
1
answer
106
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 ...
- 1,354
4
votes
0
answers
81
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 ...
- 2,461
5
votes
0
answers
128
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 ...
- 2,023
1
vote
0
answers
52
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. ...
- 2,023
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 ...
2
votes
1
answer
222
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 ...
- 393
5
votes
1
answer
211
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 ...
- 5,892
1
vote
0
answers
71
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 ...
- 3,883
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
$$\...
- 303
2
votes
1
answer
150
views
About nuclear-by-exact extensions
I know that in general exact-by-exact extensions of $C^*$-algebras need not be exact. Is it true that, if we have a short exact sequence of $C^*$-algebras
$$0 \to I \to A \to B \to 0$$
such that $I$ ...
- 105
3
votes
0
answers
255
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 ...
- 31
3
votes
1
answer
167
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{...
- 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(...
- 266
6
votes
2
answers
383
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.
...
- 10.2k
1
vote
1
answer
119
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}$ ...
- 1,135
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$-...
- 1,135
4
votes
1
answer
154
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}$ ?...
- 1,135
4
votes
0
answers
96
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$ ...
- 1,354
2
votes
1
answer
129
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 ...
- 303
4
votes
2
answers
429
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 $\...
- 2,175
11
votes
2
answers
832
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 ...
- 6,571
6
votes
0
answers
115
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$.
...
- 20.4k
4
votes
2
answers
186
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, ...
- 303
2
votes
0
answers
97
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) ...
- 953
5
votes
1
answer
303
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 ...
- 953
9
votes
0
answers
857
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 ...
- 435
3
votes
1
answer
399
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 ...
- 333
19
votes
1
answer
1k
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 ...
- 1,234
6
votes
0
answers
232
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 ...
user78249
14
votes
1
answer
784
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 ...
- 7,743
1
vote
0
answers
181
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-...
- 95
2
votes
1
answer
125
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 ...
- 501
1
vote
0
answers
160
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 ...
- 211
6
votes
1
answer
272
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 ...
- 4,040
3
votes
0
answers
289
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 ...
- 56
6
votes
1
answer
478
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) \...
- 211
6
votes
0
answers
254
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})
$$
(and there is also ...
- 349
6
votes
0
answers
257
views
Where do I read about semi-algebraic/analytic sets?
What's a good introduction to semi-algebraic/semi-analytic sets?
I'm coming from algebraic geometry, so ideally I'd prefer material that has a similar point of view and highlights analogies.
I've ...
- 1,000
15
votes
3
answers
1k
views
Can integration spoil real-analyticity?
Is there an example of a function $f:(a,b)\times(c,d)\to\mathbb{R}$, which is real analytic in its domain, integrable in the second variable, and such that the function
$$ g:(a,b)\to\mathbb{R},\qquad ...
- 343
1
vote
2
answers
143
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 ...
- 303
5
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 ...
- 349
19
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 ...
- 18.2k
1
vote
0
answers
142
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.
- 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 $g$ ...
- 303
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 ...
- 19.4k
4
votes
2
answers
424
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 ...
- 303
2
votes
0
answers
292
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 $\...
- 303
1
vote
2
answers
356
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^{*}(...
- 303