Tagged Questions

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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16
votes
1answer
487 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 ...
2
votes
2answers
322 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, ...
2
votes
1answer
118 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 ...
0
votes
1answer
219 views

Unique symplectic form in an adapted complex structure

Hallo, I ave the following question: Due to Stenzel, Lempert, Szöke ect. we know that a Riemannian manifold $(M,g)$ admits a complex structure on an neighbourhood of the cotangent bundle. This ...
1
vote
1answer
246 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? ...
4
votes
0answers
194 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. ...
2
votes
1answer
336 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 ...
1
vote
1answer
212 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 ...
2
votes
1answer
253 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 ...
3
votes
5answers
505 views

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 ...
5
votes
4answers
596 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 ...
8
votes
1answer
425 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 ...
5
votes
1answer
197 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 $G$ ...
2
votes
1answer
273 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 ...
1
vote
1answer
211 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 ...
4
votes
1answer
253 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 ...
0
votes
1answer
453 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 $M$ ...
0
votes
1answer
169 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 ...
20
votes
4answers
1k 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
3answers
549 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 ...
3
votes
1answer
545 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) ...
2
votes
1answer
733 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 ...
5
votes
1answer
368 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 ...
2
votes
1answer
186 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 $k < n$). ...
4
votes
1answer
393 views

complex manifold 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 ...
23
votes
1answer
480 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 ...
3
votes
2answers
1k 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?
1
vote
1answer
167 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 ...
0
votes
2answers
255 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 ...
2
votes
2answers
291 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 ...
3
votes
1answer
159 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 $p$. ...
12
votes
6answers
878 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 ...
16
votes
0answers
416 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 ...
-3
votes
1answer
424 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 $M$ ?
9
votes
4answers
805 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 ...
0
votes
0answers
137 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 ...
5
votes
1answer
783 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 ...
0
votes
0answers
253 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
0
votes
0answers
257 views

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

hallo, i have the following question: 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 ...
3
votes
4answers
1k 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 | ...
5
votes
1answer
484 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 ...
3
votes
2answers
629 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 ...
-1
votes
1answer
435 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 ...
4
votes
1answer
383 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 ...
2
votes
1answer
336 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 ...
0
votes
0answers
248 views

Sets that are diffeomorphic to $(0,1)^k$

Let $W\subset \mathbf R^{k}$ be an open set. Are there conditions on $W$ that guarantee the existence of a map $T:(0,1)^k \rightarrow W$ such that: (i) $T$ is surjective, (ii) $T$ is continuously ...
10
votes
1answer
505 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 ...
0
votes
1answer
225 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} = ...
5
votes
2answers
1k 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 ...
3
votes
3answers
973 views

looking for a book on banach manifolds

Hi, I am looking for a book on Banach manifolds. Can somebody recommend me something. Thanks in advance. leo