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].
1,235
questions
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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 ...
17
votes
1
answer
602
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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 ...
2
votes
2
answers
500
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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, ...
3
votes
1
answer
192
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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}(\...
1
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1
answer
499
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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. ...
1
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1
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398
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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
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0
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321
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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. ...
4
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1
answer
1k
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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 ...
1
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1
answer
544
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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 ...
2
votes
1
answer
402
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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 $\...
8
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5
answers
2k
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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 \...
9
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3
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2k
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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 ...
9
votes
1
answer
615
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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 ...
6
votes
1
answer
460
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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 $G$ ...
2
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1
answer
404
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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....
3
votes
1
answer
532
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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 (...
4
votes
1
answer
355
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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 ...
0
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1
answer
608
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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 $M$ ,...
0
votes
1
answer
193
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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 ...
25
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4
answers
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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
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3
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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 $h:P\rightarrow\mathbb{R}$...
3
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1
answer
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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) ...
8
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1
answer
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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 ...
5
votes
1
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448
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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 ...
3
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1
answer
248
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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 $k < n$). ...
7
votes
1
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701
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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 ...
25
votes
1
answer
697
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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 ...
5
votes
1
answer
5k
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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?
2
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1
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222
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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 ...
0
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2
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1k
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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-...
2
votes
2
answers
322
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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?...
3
votes
1
answer
253
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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 $p$. ...
22
votes
6
answers
3k
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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 ...
24
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0
answers
1k
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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 ...
-2
votes
1
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871
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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 $M$ ?
14
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4
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3k
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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 $...
1
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0
answers
165
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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 ...
5
votes
2
answers
691
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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 ...
6
votes
1
answer
2k
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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 $f$...
1
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0
answers
300
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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
2
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0
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445
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$\partial \bar{\partial}$ 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 ...
4
votes
4
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2k
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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 | \...
5
votes
1
answer
799
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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 ...
5
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2
answers
2k
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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 $...
0
votes
1
answer
481
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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 (...
4
votes
1
answer
804
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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 ...
3
votes
1
answer
765
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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 ...
11
votes
1
answer
873
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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 ...
1
vote
1
answer
251
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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. $d\omega_{i} = 0$)...
17
votes
3
answers
5k
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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 ...