Questions tagged [smoothness]

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83 votes
4 answers
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Parallelizability of the Milnor's exotic spheres in dimension 7

Are the Milnor's seven dimensional exotic spheres parallelizable?
Hamed's user avatar
  • 1,226
21 votes
1 answer
1k views

A differentiable isometry is smooth?

I posted this question in MSE but got no response (even after giving a bounty), so I am trying here. Let $M,N$ be smooth $d$-dimensional Riemannian manifolds. Suppose $f:M \to N$ is a differentiable ...
Asaf Shachar's user avatar
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20 votes
5 answers
2k views

Smoothness of the closest point on a submanifold

Let $(M,g)$ be a smooth Riemannian manifold, and let $S \subseteq M$ be a compact submanifold. Assume that for each $p \in M$, there exist a unique closest point on $S$, i.e a unique point $\tilde s(...
Asaf Shachar's user avatar
  • 6,611
19 votes
4 answers
2k views

Example of a smooth morphism where you can't lift a map from a nilpotent thickening?

Definition. A locally finitely presented morphism of schemes $f\colon X\to Y$ is smooth (resp. unramified, resp. étale) if for any affine scheme $T$, any closed subscheme $T_0$ defined by a square ...
Anton Geraschenko's user avatar
18 votes
3 answers
2k views

Is there an example of a variety over the complex numbers with no embedding into a smooth variety?

Is there an example of a variety over the complex numbers with no embedding into a smooth variety?
David Zureick-Brown's user avatar
17 votes
2 answers
935 views

Kolmogorov superposition for smooth functions

Kolmogorov superposition theorem states that a continuous function $f(x_1,\ldots,x_n)$ can be written as $$f(x_1,\ldots,x_n)=\sum_{q=0}^{2n}\Phi_q\left(\sum_{p=1}^{n}\phi_{q,p}(x_p)\right)$$ for ...
O.R.'s user avatar
  • 807
16 votes
2 answers
3k views

Is there an example of a formally smooth morphism which is not smooth?

A morphism of schemes is formally smooth and locally of finite presentation iff it is smooth. What happens if we drop the finitely presented hypothesis? Of course, locally of finite presentation is ...
David Zureick-Brown's user avatar
15 votes
1 answer
2k views

Interpolating between piecewise linear functions, with a family of smooth functions

Let $[a,b)\subset\mathbb R$, and $F,G:[a,b)\to\mathbb R$ two decreasing piecewise linear functions so that $F(x)\leq G(x)$ for any $x\in[a,b)$. We assume that: there is a number $k\in\mathbb N-\{0\}$ ...
Cristi Stoica's user avatar
15 votes
0 answers
310 views

How much smoothness does the tennis ball theorem need?

The tennis ball theorem states that a smooth-enough curve that bisects the surface area of a sphere must have at least four inflection points. There are plenty of sources on this but most of them are ...
David Eppstein's user avatar
14 votes
2 answers
3k views

Slice knots and exotic $\mathbb R^4$

In the http://arxiv.org/abs/math/0606464v1 I read "If you want to prove existence of exotic smooth structure on $\mathbb R^4$ you can do this if you are in possession of a knot which is ...
Nikita Kalinin's user avatar
14 votes
0 answers
542 views

(When) is isomorphism on differentials enough to guarantee that a map is étale?

I'm sorry if this is too easy for MO. Let $S$ be a locally noetherian scheme, flat over $\mathrm{Spec}\,\mathbb{Z}$, $X$ and $Y$ be flat $S$-schemes locally of finite presentation, and let $f:X\to Y$ ...
Piotr Achinger's user avatar
12 votes
1 answer
382 views

Are algebras of smooth functions formally smooth?

Let $M$ be a manifold. Then is the ring of smooth functions $C^\infty(M,\mathbb{R})$ formally smooth over $\mathbb{R}$? If it helps, feel free to assume that $M$ is compact. (This is not a joke ...
Tobias Fritz's user avatar
  • 5,775
12 votes
2 answers
648 views

Is the square root of a monotonic function whose all derivatives vanish smooth?

Let $g:[0,\infty] \to [0,\infty]$ be a smooth strictly increasing function satisfying $g(0)=0$ and $g^{(k)}(0)=0$ for every natural $k$. Is $\sqrt g$ is infinitely (right) differentiable at $x=0$? ...
Asaf Shachar's user avatar
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12 votes
0 answers
479 views

Exotic smoothness and Parallelizability

Regarding the parallelizability of the Milnor's seven dimensional exotic spheres: Parallelizability of the Milnor's exotic spheres in dimension 7 The following question naturally arises: Suppose ...
Hamed's user avatar
  • 1,226
11 votes
2 answers
624 views

Do locally convex topological vector spaces embed into diffeological spaces?

The nLab casually remarks that locally convex tvs embed into diffeological spaces by (discussion around) a corollary in Kriegl and Michor, namely 3.14, but this deals with Boman's theorem and results ...
David Roberts's user avatar
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11 votes
3 answers
609 views

smooth functional to detect whether a function has a zero

Does there exist a function $F : C^\infty(\mathbb{R}, [0, \infty)) \to \mathbb{R}$ with the following properties: $F(f) = 0$ if and only if there exists an $x \in [0,1]$ such that $f(x) = 0$. $F$ is ...
Dan Christensen's user avatar
11 votes
2 answers
1k views

Do smooth ind schemes have Dualizing sheafs?

Say I have an ind scheme $X = \cup_i X_i$ over a field $k$. I have its tangent bundle $\hom_k(k[\epsilon], X)$ which I can think of as ind scheme via $\cup_i \hom_k(k[\epsilon],X_i)$. The problem is ...
solbap's user avatar
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9 votes
3 answers
1k views

If $\Omega_{X/Y}$ is locally free of rank $\mathrm{dim}\left(X\right)-\mathrm{dim}\left(Y\right)$, is $X\rightarrow Y$ smooth?

Suppose I have a morphism $f:X\rightarrow Y$ such that the relative sheaf of differentials $\Omega_{X/Y}$ is locally free. Does it follow that $f$ is smooth? The answer is no, but for a silly reason. ...
Anton Geraschenko's user avatar
9 votes
1 answer
939 views

on the local structure of schemes

Let $X$ be an integral finite type scheme over $\mathbb{C}$. Let $x\in X$, such that there exists a neighborhood $U$ of $x$, such that the sheaf of differentials $\Omega^{1}_{U}$ decomposes into: $\...
prochet's user avatar
  • 3,432
9 votes
1 answer
514 views

Can one check formal smoothness using only one-variable Artin rings?

Let $f:X\rightarrow Y$ be a morphism of schemes over a field $k$. Can one check that $f$ is formally smooth using only Artin rings of the form $k^{\prime}\left[t\right]/t^{n}$, where $k^{\prime}$ is ...
David Zureick-Brown's user avatar
8 votes
2 answers
333 views

On Glaeser's Theorem for non-smooth functions

Glaeser's Theorem says that a $C^\infty$ function $F$ on $\mathbb R^n$ which is invariant under permutation of the variables is a smooth function of the symmetric polynomials of $(x_1, \dots, x_n)$. ...
Bazin's user avatar
  • 15.1k
8 votes
2 answers
349 views

Can smoothness of curves into a convenient locally convex vector space be tested with just a dense subspace of the dual?

Let $E$ be a (Hausdorff) locally convex vector space (from now on just "lcs" for short). We say that $E$ is convenient (also called locally complete, Mackey-complete or $c^\infty$-complete) if, given ...
Pedro Lauridsen Ribeiro's user avatar
8 votes
0 answers
340 views

Interpretation of $p$-adic 'smoothness'

real case: In the very first course of Calculus, one learns that a real function $f \colon \mathbb{R} \to \mathbb{R}$ is called smooth, if it is differentiable as many times as one pleases. So the ...
Maty Mangoo's user avatar
8 votes
0 answers
328 views

Criterion for smooth functions [duplicate]

Let $f:\mathbb{R}→\mathbb{R}$ a real-valued function and $m,n∈\mathbb{N}^∗$ coprime, i.e. greatest common divisor of m and n is 1, and define $f^m:=f\cdot f\cdot\ldots\cdot f.$ Show that $$f^m,f^n\in ...
stefano's user avatar
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8 votes
0 answers
458 views

How should I try to imagine exotic smoothness in R4?

I am trying to wrap my mind around the concept of exotic smoothness in (and only in) $\mathbb{R}^4$. I have some mathematical literature, but can anyone point to a semi-intuitive, semi-visual example?...
Giulio Prisco's user avatar
7 votes
4 answers
1k views

How singular can the Stein factorization of a proper map between smooth varieties be?

A little bit of motivation (the question starts below the line): I am studying a proper, generically finite map of varieties $X \to Y$, with $X$ and $Y$ smooth. Since the map is proper, we can use the ...
jmc's user avatar
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7 votes
1 answer
817 views

Bertini's theorem over non-algebraically closed field

Let $K$ be a non-algebraically closed (infinite) field of characteristic $0$ and $X$ a smooth, projective $K$-variety. Does there exist an ample invertible sheaf $\mathcal{L}$ on $X$ such that a ...
user43198's user avatar
  • 1,949
7 votes
1 answer
217 views

Is the Borel lemma projection a smooth principal bundle?

Consider the Fréchet spaces $C^\infty(\mathbb{R},\mathbb{R})$ and $\mathbb{R}^\infty$, and the continuous linear map $$ J\colon C^\infty(\mathbb{R},\mathbb{R}) \to \mathbb{R}^\infty $$ returning the ...
David Roberts's user avatar
  • 33.8k
7 votes
1 answer
530 views

Smoothness of a projective variety via the derived category

Let $X$ be a smooth projective integral variety over an algebraically closed field $k$. Let $Y$ be a (not necessarily smooth) projective integral variety over $k$. Assume that $D^b(X) \cong D^b(Y)$. ...
Marco's user avatar
  • 71
7 votes
2 answers
589 views

Is $(x^2y,xy^2)$ log smooth?

Consider the map $$f:\mathbb C^2\to\mathbb C^2$$ $$(x,y)\mapsto(x^2y,xy^2)$$ We can view $f$ as induced by the map of monoids $g:\mathbb Z^2_{\geq 0}\to\mathbb Z^2_{\geq 0}$ given by the matrix $(\...
John Pardon's user avatar
  • 18.3k
7 votes
1 answer
367 views

Are metric isometries smooth at the boundary?

Let $M,N$ be smooth Riemannian manifolds with boundary (In particular, we assume the boundaries are smooth). Suppose we have a map $\phi:M \to N$ which satisfies the following properties: $$(1) \, \,...
Asaf Shachar's user avatar
  • 6,611
7 votes
0 answers
288 views

Smooth dependence on parameters of invariant manifolds

This may be considered a followup question to smooth dependence of stable manifold on parameters. In particular, I would like to know in more detail how smooth parameter dependence of a (generalized) ...
Jaap Eldering's user avatar
6 votes
2 answers
1k views

Zero points of a smooth function on $\mathbb{R}$

Assume $f(x)$ is a smooth function on $\mathbb{R}$ and $f$ does not vanish on any interval. In other words, $f$ can have zero points but we cannot find any interval $(a, b)$ such that $f(x)=0$ for all ...
Jacob Lu's user avatar
  • 903
6 votes
2 answers
314 views

Path algebras are formally smooth

In Ginzburg's notes Lectures on Noncommutative Geometry, he claim that the path algebra of a quiver is formally smooth (See Section 19.2). I have two questions. First, how to show this claim and ...
Yining Zhang's user avatar
6 votes
5 answers
2k views

Does smooth target space and smooth fibers imply smooth total space?

Suppose that $f: X \rightarrow Y$ is a morphism between algebraic varieties. If $Y$ is smooth, and the fibers of $f$ over closed points of $Y$ are proper and nonsingular, does it follow that $X$ is ...
unknown's user avatar
  • 181
6 votes
1 answer
902 views

Is every variety an image of a smooth variety?

Let $X$ be a finite type scheme over a field $k$. Is it true that there exists a surjective morphism $f : Y \rightarrow X$, where $Y$ is smooth over $k$? In other words, is every such scheme a ...
P. Grabowski's user avatar
6 votes
2 answers
328 views

A smooth function such that the second derivative of its absolute value is a distribution of positive order

Let $f\in C^\infty(\mathbb R;\mathbb R)$ and let us define $g(x)=\vert f(x)\vert$. It is easy to verify that $g$ is locally Lipschitz-continuous function, but I would like to find an example of a ...
Bazin's user avatar
  • 15.1k
6 votes
1 answer
108 views

$C^{k, \alpha}$ gradients $\implies C^{k + 1, \alpha}$ level sets

Referring to the statement in the parentheses below (Regularity Theory for Elliptic PDE, Xavier Ros-Oton, p. 172) -- the authors go forward to make a bootstrapping argument that uses this result: ...
user43389's user avatar
  • 245
6 votes
1 answer
451 views

A smooth function $\mathbb{R}\to\mathbb{R}$ agrees with an analytic function on a bounded infinite set

Fix a smooth function $f:\mathbb{R}\to\mathbb{R}$. Do there exist real numbers $a<b$, an infinite set $S\subset (a, b)$ and an analytic function $g$ defined on $(a-\epsilon, b+\epsilon)$ for some $\...
user avatar
6 votes
1 answer
287 views

On Glaeser's result for the square-root of a smooth non-negative function

One of the results due to Georges Glaeser is the following: there exists a non-negative $C^\infty$ function $f$ on the real line, flat at its zeroes, such that $\sqrt{f}$ is not $C^2$. On the other ...
Bazin's user avatar
  • 15.1k
6 votes
0 answers
230 views

Unitary representations of finite dimensional Lie groups on infinite-dimensional Hilbert spaces

I am interested in the proofs of continuity of some standard unitary representations appearing in Physics. Additionally, I am interested in the integration of finite-dimensional Lie algebras of skew-...
Javier's user avatar
  • 483
5 votes
2 answers
1k views

Derivatives of $C^{\infty}$ non analytic function

Question: Given $f\in C^{\infty}$ which is not analytic on a bounded domain $\Omega \subseteq \mathbb{R}$. What can we say about the sequence $\lbrace f^{(m)} \rbrace _{m=1}^{\infty} $? Specifically - ...
Amir Sagiv's user avatar
  • 3,544
5 votes
1 answer
656 views

A regular, geometrically reduced but non-smooth curve

Can anyone give an example of a projective, regular, geometrically reduced but non-smooth curve ? Of course, the base field should be imperfect. In Exercise 4.3.22 of Qing Liu's book Algebraic ...
Yong Hu's user avatar
  • 610
5 votes
2 answers
181 views

Differentiability of polytope shadow areas

Let $P$ be an opaque convex polyhedron containing the origin in $\mathbb{R}^3$, and let $S$ be an origin-centered sphere strictly containing $P$: $S \supset P$. For a point $x$ on $S$, let $\sigma(x)$ ...
Joseph O'Rourke's user avatar
5 votes
4 answers
584 views

Relative version of Hilbert syzygy theorem

I presume that answers to the following questions are likely to exist in the literature; so this question is mostly a reference request (but failing that, I would be certainly interested in learning a ...
Leonid Positselski's user avatar
5 votes
1 answer
343 views

Extension of functions from geodesically convex compact sets in a Riemannian manifold

In the paper Extension operators for spaces of infinite differentiable Whitney jets (J. reine angew. Math. 602 (2007), 123—154, DOI:10.1515/crelle.2007.005) by Leonhard Frerick, a convenient condition ...
David Roberts's user avatar
  • 33.8k
5 votes
1 answer
263 views

Smoothness of the radius of convergence

Let $(x\mapsto a_n(x))_n$ be a sequence of smooth functions defined on some fixed interval $I$. Consider the power series $\sum_{n\geq 0}a_n(x)t^n$ and denote by $R(x)$ its radius of convergence. Does ...
M. Dus's user avatar
  • 1,900
5 votes
1 answer
395 views

Confusion with formally unramified = immersion and formally smooth = submersion

From this MO question I learned to tentatively think of formally unramified arrows as immersions and of formally smooth arrows as submersions. I'm trying to semi-formally handwave myself into ...
Arrow's user avatar
  • 10.3k
5 votes
1 answer
162 views

Critical Smoothness on Besov Spaces $B^s_{p}$: how does it evolved with $p$?

We denote by $B_{p}^s(\mathbb{T}) := B_{p,p}^s(\mathbb{T})$ the Besov space over the circle $\mathbb{T}$ with parameters $p=q \in (0, \infty]$ and smoothness $s \in \mathbb{R}$. For $p>0$ fixed and ...
Goulifet's user avatar
  • 2,174
4 votes
1 answer
513 views

Uniqueness of smooth compactification upto a smooth morphism

By a $k$-variety, we will mean a separated scheme of finte type over a field $k$. Let $k$ be of characteristic 0. Given a smooth quasi-projective $k$-variety $X$, there is a projective $k$-variety $\...
Anandam Banerjee's user avatar