# Questions tagged [real-analytic-structures]

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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 ...
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-...
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 ...
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 ...
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 ...
1 vote
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. ...
1 vote
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 ...
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 ...
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 ...
1 vote
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 ...
148 views

1 vote
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 ...
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 ...
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 ...
1 vote
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.
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 $g$ ...
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 ...
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 ...