Questions tagged [smooth-manifolds]

Smooth manifolds and smooth functions between them. For manifolds with additional structure, see more specific tags, such as [riemannian-geometry]. For more topological aspects, see [differential-topology].

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Geodesics on a twisted torus

This is a repost of a question I posted at MSE. Mark L. Irons' paper The Curvature and Geodesics of the Torus gives a concise overview of the geodesics on the torus: There are five clear-cut ...
Hans-Peter Stricker's user avatar
17 votes
1 answer
602 views

Is there a notion of a chain complex with corners?

Roughly speaking, algebraic topology works by reducing questions about topological objects such as manifolds and cell to questions about chain complexes. On the topological side, although in the PL ...
Daniel Moskovich's user avatar
2 votes
2 answers
500 views

Isometric embedding of a Kaehler manifold as a special Lagrangian in a Calabi-Yau manifold

Hallo, I am reading the paper "Hyperkaehler structures on the total space of holomorphic cotangent bundles" by D.Kaledin and I am asking if it is possible to embedd a real-analytic Kähler manifold, ...
Pavel's user avatar
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3 votes
1 answer
192 views

Can stabilizer groups in an orbifold have global twisting?

Can stabilizer groups in an orbifold have global twisting? For example, consider the two groups $\mathbb Z/3\times\mathbb Z$ and $\mathbb Z/3\rtimes\mathbb Z$ (where $\mathbb Z\to\operatorname{Aut}(\...
John Pardon's user avatar
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1 vote
1 answer
499 views

Unique symplectic form in an adapted complex structure

I have the following question: Due to Stenzel, Lempert, Szöke etc. we know that a Riemannian manifold $(M,g)$ admits a complex structure on a neighbourhood of the zero section of the cotangent bundle. ...
Joan's user avatar
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1 vote
1 answer
398 views

How can I picture antisymmetry of the Lie derivative?

It's obvious that the Lie derivative defined in terms of Lie brackets is anti-symmetric. But what is an intuitive way to visualize the anti-symmetry in the 'differentiating along a flow' definition? ...
6 votes
0 answers
321 views

Ricci-flat metrics on Cotangent bundles in adapted complex structure

greetings, Let $(M,g)$ be a compact Riemannian manifold. On some neighbourhood $X$ of the zero section in the cotangent bundle $T^{*}L$ we have a complex structure $J$ and a Kähler form $\omega$ s.t. ...
dominik's user avatar
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4 votes
1 answer
1k views

Definition of Sobolev spaces as a space of sections of certain type

I want to define Sobolev spaces for sections on a vector bundle, basically I want that a section will belong to the Sobolev space $W^{k,p}$ if its coordinates in any aceptable patch belong to the ...
inquisitor's user avatar
1 vote
1 answer
544 views

Kähler manifold with Ricci-flat Kähler form

hallo, I have the following problem: Let $X$ be a $n-$dim Kähler manifold with Ricci-flat Kähler form $\omega$. There is a known fact that then there exists a holomorphic (n,0)-form $\Omega$ such ...
bruno's user avatar
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2 votes
1 answer
402 views

holomorphic extension of forms

hallo, I have the following question: Let $M$ be a $n-$dimensional complex manifold and $X \subset M$ be a compact $n-$dimensional totally real analytic Riemannian submanifold. Let furthermore $\...
bruno's user avatar
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8 votes
5 answers
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Examples of manifolds with effective circle actions?

I would like to know examples of smooth compact connected manifolds, on which there exists an effective smooth circle action preserving a positive smooth volume, besides the simple example: $[0,1]^d \...
Mostafa's user avatar
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9 votes
3 answers
2k views

When does a hypersurface have contact-type?

In a symplectic manifold $(X^{2n},\omega)$, a hypersurface $Y\subset X$ has contact-type if there is a contact form $\lambda$ such that $d\lambda=\omega|_Y$. Recall that a contact form is a 1-form ...
Chris Gerig's user avatar
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9 votes
1 answer
615 views

Exponentiable objects in a category, valued in a larger, containing category

Recall that when dealing with topological spaces one usually likes dealing with a subcategory of $Top$ which is convenient, one facet of which is that it is cartesian closed. However to get to a ...
David Roberts's user avatar
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6 votes
1 answer
460 views

Equivariant handle decompositions

Suppose I have some smooth closed high-dimensional manifold $M$ acted on smoothly by a finite group $G$. By a metric averaging procedure, we can equip $M$ with a smooth Riemannian metric so that $G$ ...
John Pardon's user avatar
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2 votes
1 answer
404 views

smooth manifold vs. exceptional inverse image

A well-known theorem in topology says that for a smooth manifold $M$ of dimension $n$ the map $f: M \rightarrow point$ satisfies $$f^! \mathbf R = \mathbf R[n]$$ Here $\mathbf R$ is the constant sheaf....
Jakob's user avatar
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3 votes
1 answer
532 views

Harmonic/conformal map composition between manifolds in either order?

Suppose $\mathcal{M}$, $\mathcal{N}$, and $\mathcal{P}$ are Riemannian manifolds (compact and of dimension 2, if it matters). It seems well-known that if $\phi:\mathcal{M}\rightarrow\mathcal{N}$ is (...
Justin's user avatar
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4 votes
1 answer
355 views

Classifying smooth embeddings which yield Morse functions

Let $\mu:M \to \mathbb{R}$ be a fixed surjective smooth function on a smooth manifold $M$. Let $N$ be a smooth compact manifold that embeds smoothly into $M$ via $\iota:N \to M$. What conditions on ...
Vidit Nanda's user avatar
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0 votes
1 answer
608 views

sign of the First chern class fundamental group of Kahler Manifolds

We know by some facts from Kobayashi, if the Kahler manifold $M$ has positive first Chern class, i.e., $c_1 (M)>0$ then $M$ is simply connected. So if $c_1 (M)<0$ under which assumption on $M$ ,...
user avatar
0 votes
1 answer
193 views

relation with jacobifields in a small neighbourhood

hi, I have the following question: Let $(M,g)$ be a complete Riemannian manifold with all sectional curvatures non-positive. Let $p \in M$ and consider the function $d(x)=dist_{g}(x,p)$ in a ...
pascal's user avatar
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25 votes
4 answers
3k views

Algorithmically unsolvable problems in topology

This question is inspired by a paper by B. Poonen that appeared on the arxiv some time ago: http://arxiv.org/abs/1204.0299. The paper gives a sample of algorithmically unsolvable problems from various ...
4 votes
3 answers
1k views

When do commuting Hamiltonian flows have commuting generators?

Let $(P,\Omega)$ be a symplectic manifold, and let $[\cdot,\cdot]$ be the natural Poisson bracket. Let $\varphi^h(a)$ be the Hamiltonian flow generated by the smooth function $h:P\rightarrow\mathbb{R}$...
soulphysics's user avatar
3 votes
1 answer
1k views

Homeomorphism classification of 4-manifolds

Question 1. Let $X_i$ be an infinite family of closed, orientable, smooth 4-manifolds with the following properties: a) $\pi_1(X_i) = \mathbb{Z}\times \mathbb{Z_{2}}$ for any $i = 1, 2, \cdots $ b) ...
user23802's user avatar
8 votes
1 answer
5k views

Manifolds are paracompact

By Definition, smooth manifolds are assumed to be Hausdorff and to satisfy the second countability axiom. I have heard (but never seen written) that these assumptions imply paracompactness (and thus ...
ThiKu's user avatar
  • 10.3k
5 votes
1 answer
448 views

Actions of finite groups on exotic smooth manifolds of dimension >4

Let $M_1^n$ and $M_2^n$, $n>4$ be two smooth compact manifolds that are homeomorphic but not diffeomorphic. Suppose that a finite group is $G$ acting faithfully on $M_1^n$ by diffeomorphisms. Is it ...
aglearner's user avatar
  • 14k
3 votes
1 answer
248 views

gluing along a real analytic manifold

hi, I have a general question. Assume we have a real analytic $n-$dim. manifold $X$ and $M$ a real analytic compact submanifold of $X$ (of dimension less that the dimension of $X$, say $k < n$). ...
pascal's user avatar
  • 89
7 votes
1 answer
701 views

Complex manifolds with corner?

I was reading Dominic Joycee article on Manifold with corner. He talk about manifold with corner modeled over $[0,\infty)^k\times \mathbb R^{n-k}$ for some $k\leq n$. From here I moved to Melrose ...
Jonujohn's user avatar
  • 217
25 votes
1 answer
697 views

Diffeomorphisms of finite order not in the image of a circle action

Does there exist a closed smooth manifold $M$ and a diffeomorphism $f\colon M \to M$ such that: $f$ is isotopic to the identity, $f$ is of finite order, $f^n=ID$, and $f$ is not contained in the ...
Jarek Kędra's user avatar
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5 votes
1 answer
5k views

What is the usual topology of $C^\infty_c(M) $

If $M$ is a smooth paracompact manifold, then what is the usual topology of $C^\infty_c(M) $, i.e., the smooth function with compact support?
Adterram's user avatar
  • 1,361
2 votes
1 answer
222 views

Morse Theory on pseudo-Hermitian manifold

I wonder if Morse Theory on pseudo-Hermitian manifold is developed. For example, I wonder if the following statement on pseudo-Hermitian manifold, which is corresponding to the Riemannian case, is ...
Paul's user avatar
  • 834
0 votes
2 answers
1k views

questions on intersecting 2-manifolds

Suppose two intersecting smooth manifolds which are both subset of $\mathbb{R}^2$, and their tangent spaces on points of the intersecting parts doesn't coincident. Then is this intersecting part a 1-...
user23153's user avatar
2 votes
2 answers
322 views

Questions on calculating volume using n-1 forms

Is there an n-1 form on $R^n$ which calculates the volume of n-manifolds? Similarly, is there such a 1 form on $S^2$, and $RP^2$? I thought this has something to do with the orientation, is that right?...
user23153's user avatar
3 votes
1 answer
253 views

Boundary of unstable manifold

Let $X$ be a vector field on a compact manifold $M$ that has the form $$ X = \lambda_1 x^1 \partial_1 + \dots + \lambda_n x^n \partial_n + \dots$$ with respect to some chart $x$ around a point $p$. ...
Matthias Ludewig's user avatar
22 votes
6 answers
3k views

Does every vector bundle allow a finite trivialization cover?

Suppose there is a vector bundle (smooth, with constant rank finite-dimensional fibres) over a (smooth, second-countable, Hausdorff, not necessarily connected) manifold $B$ of dimension $n$. (a) Is ...
Fiktor's user avatar
  • 1,264
24 votes
0 answers
1k views

Monoid structure of oriented manifolds with connect sum

Take the class of all compact, connected, boundaryless, smooth oriented $n$-dimensional manifolds, each taken up to orientation-preserving diffeomorphism. This is a commutative monoid with operation ...
Ryan Budney's user avatar
  • 42.8k
-2 votes
1 answer
871 views

Holonomy group of calabi yau manifold

Let $(M,J,\omega, \Omega)$ be a calabi-yau manifold (not necessary compact). Does it follow that the holonomy group of $M$ is $SU_{n}$, where $n$ is the complex dimension of $M$ ?
pascal's user avatar
  • 89
14 votes
4 answers
3k views

Characterization of the Lie derivative

The exterior differential of differential forms on a manifold can be characterized as the unique super-derivation of degree 1 on the exterior algebra of forms such that $<df,X>=X(f)$ for $f$ a $...
A. Pascal's user avatar
  • 1,309
1 vote
0 answers
165 views

monge ampere equation along totally real submanifolds

hi, are there some references when solving the complex monge ampere equation along totally real submanifolds of some compact (with boundary or without) complex manifold. i know that there are a lot ...
william's user avatar
  • 213
5 votes
2 answers
691 views

Does a manifold which bounds always admit a free involution?

If a closed smooth manifold $M$ admits a smooth free involution $T$, then it bounds. In fact, the mapping cylinder of the quotient map $M \to M/T$ is the manifold whose boundary is $M$. Is the ...
kelly's user avatar
  • 127
6 votes
1 answer
2k views

Classification of smooth atlases

Let $\mathcal{A}$ be a smooth maximal atlas on a manifold $M$. Let $f:M\to M$ be a smooth invertible function, whose inverse is not smooth (for example $f:\mathbb R\to \mathbb R$, $f(x)=x^3$). Then $f$...
Cristi Stoica's user avatar
1 vote
0 answers
300 views

einstein metrics on the tangent bundle

hi, i have the following question. let $M$ be a compact, real analytic, riemannian manifold with real analytic metric $g$. does the tangent bundle admit always a einstein metric ? marco
william's user avatar
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2 votes
0 answers
445 views

$\partial \bar{\partial}$ on a complex manifold

Let $M$ be a complex $n$-dimensional manifold and $R \subset M$ be a totally real, compact, $n$-dimensional (real) manifold. Let $\alpha$ be a smooth nonnegative $(n,n)-$form on $M$. Does there exist ...
william's user avatar
  • 213
4 votes
4 answers
2k views

space of geodesics

hallo, i have the following problem: Let $(M,g)$ be a compact Riemannian manifold with metric $g$ and $\nabla$ be the Levi-Civita Connection. Denote by $G(M) =${$\gamma: \mathbb{R} \rightarrow M | \...
william's user avatar
  • 213
5 votes
1 answer
799 views

Partitions of Unity

Fix a metric $g$ on a smooth, closed manifold $\mathcal{M}$. Take a finite subcover of the manifold from its atlas. Is it true that any smooth partition of unity subordinate to this cover has ...
Michael Coffey's user avatar
5 votes
2 answers
2k views

definition of Hessian with respect to connection

Hi, I am reading the lecture notes on Morse Homology written by M.Hutchings, in that notes definition of Hessian is defined in coordinate free way: given any connection $ H(f,p)= \nabla_v(df)$ where $...
zatilokum's user avatar
  • 205
0 votes
1 answer
481 views

Is this manifold orientable? [closed]

Let $C$ be the set of points $(a,b,c,d) \in \mathbb{C}^4$ which satisfy 1) $ \left|a\right|^2+\left|c\right|^2=\left|b\right|^2+\left|d\right|^2 =1 $. 2) $ a\bar{b}+c\bar{d}=0 $ There is a (...
Mohammad Farajzadeh-Tehrani's user avatar
4 votes
1 answer
804 views

Is the space of smooth partitions of unity connected? Simply-connected?

One of the requirements for a smooth manifold $M$ is that it be paracompact, and one of the equivalent definitions of paracompactness for a smooth space is that for overy open cover of $M$, there ...
Daniel Moskovich's user avatar
3 votes
1 answer
765 views

Conformally-flat

Assume given a smooth manifold $(\mathbb{R}^n, g)$, where the metric is a scaled identity $g = e^{2f}I$. Is there a way to know if this is always a non-positive (sectional) curvature manifold? Note ...
Guillermozo's user avatar
11 votes
1 answer
873 views

Smooth four-manifolds with contractible universal cover

Let $X$ be a smooth compact four-manifold with definite non-trivial intersection form. Can the universal cover of $X$ be contractible? It semms to me that the answer is negative when $X$ is simply ...
Matei's user avatar
  • 123
1 vote
1 answer
251 views

local kählerforms on complex manifold

hallo, Let $M$ be a complex manifold. Assume we have a covering of $M$ by complex charts $\{U_{i}\}$. Furthermore assume that we have on each $U_{i}$ a Kählerform $\omega_{i}$ (i.e. $d\omega_{i} = 0$)...
gary's user avatar
  • 221
17 votes
3 answers
5k views

one-parameter subgroup and geodesics on Lie group

Hi, Given a Matrix Lie Group, I would like to know if the one-parameter subgroups (which can be written as $\exp^{tX}$) are the same as the geodesics (locally distance minimizing curves). Geodesics ...
frank's user avatar
  • 173