# 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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### Intuition behind the following theorem of Reeb?

What is the intuition behind the following theorem of Reeb? If a compact manifold admits a function with only two critical points which are non degenerate, it is homeomorphic to the sphere.
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### Submersion theorem for smooth tame Frechet manifolds

If $M$ and $N$ are Banach manifolds, $f:M\rightarrow N$ is a smooth map, and $q\in N$ is a regular value, so $f$ is a submersion on $f^{-1}(q)$, it is well known that the level set $f^{-1}(q)$ is a ...
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### Stokes-like Theorem for Dolbeault Operator

I have a simple question regarding complex geometry: is there an analog for the Stokes Theorem for the Dolbeault Operator $\bar{\partial}$? For instance, suppose that $M$ is a closed complex manifold ...
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### Suitable reference on smooth manifolds for qualifying exam study?

Is there a single suitable reference to study for the smooth manifold (geometry) half of a typical Topology/Geometry PhD preliminary exam at an average AMS group I school? (Note: I know details about ...
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### Poisson algebra automorphisms of a symplectic manifold

Let $(M,\omega)$ be a symplectic manifold. Let $V=\mathcal{C}^\infty(M)$ be the Lie algebra of smooth real valued functions. Suppose $f:\rightarrow V$ be an Lie-algebra isomorphism (an algebra ...
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### Can the Laplace operator on $n-$ manifolds be represented as a sum of $n$ second order derivational operators

EDIT: According to some comments on this post I revise the title to remove the misunderestanding. Assume that $M$ is a Riemannian manifold of dimension $n$. The natural Laplace operator associated ...
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### Smooth manifolds for which every metric is geodesically convex

Are there non compact smooth manifolds which have the property that every Riemannian metric is geodesically convex? Note that a manifold for which every Riemannian metric is complete must be compact. ...
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### How close (Homology-wise) can we approximate a topological manifold with a PL or smooth one?

Sorry if this question is to naive or badly phrased. I am curious about the following problem, given a manifold $M$, how "close" can we find a smooth or PL manifold, $N$, with a map $f:N\to M$. The ...
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### Tube formula for r-neighbourhood of a manifold

Let $P$ be a topologically embedded submanifold in a Riemannian manifold $M$. Then the tube $T(P, r)$ of radius $r \geq 0$ about $P$ is the set of all points $m \in M$ such that there exists a ...
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### Closed orientable 4-manifold with $H^1(M;\Bbb Z_2)=\Bbb Z_2$ and non-zero cup product $H^1\times H^1\to H^2$

I am looking for an example of a closed orientable 4-manifold $M$ with $H^1(M;\Bbb Z_2)=\Bbb Z_2$ and non-zero cup product $H^1(M;\Bbb Z_2)\times H^1(M;\Bbb Z_2)\to H^2(M;\Bbb Z_2)$. A non-orientable ...
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### Certain principal bundle structure on $\mathbb{R}^{n} \setminus \{0\}$

I ask this question in MSE and I received no answer, so I repeat it here: Is there a right action of $\mathbb{H}^{2}$ on some $\mathbb{R}^{n}\setminus \{0\}$ such that this action gives us a ...
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### Taylor expansion in Riemannian foliations

Take: $M$ a Riemannian manifold, ${X_0}\in M$, $N_{X_0}$ a submanifold of $M$ going through ${X_0}$, and $Z \in N_{X_0}$ in a neighborhood of ${X_0}$. At ${X_0} \in N_{X_0}$, we consider the ...
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### Symplectic reversing diffeomorphisms on a compact symplectic manifold

I Ask this question in MSE and I received interesting comments and ideas. I repeat the question here for more discussion: Let $(M,\omega)$ be a compact symplectic manifold. Is there a ...
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### Exponential map and convergence [closed]

I posted this question on Math Stack Exchange, but nobody answered so I decided to ask this question here. Suppose that $M$ is smooth compact manifold and let $y \to x$. Let also $f \in C^{\infty}(M)$ ...
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### Differential Topology over $\mathbb{Q}$

I have a specific question in mind, but it requires some explanation and context before it can be formally stated. To summarize it in a sentence, this is it: Are every two rational manifolds of the ...
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I am trying to derive the auto-parallel equation for a curve. Consider a smooth manifold $M$ with chart map $x$, and let $\gamma$ be a smooth curve in $M$. The action of its tangent vector $v_{\gamma}... 1answer 58 views ### Glueing smooth functions give a smooth function if reparametrized [closed] Given$\mathbf{r}:(t_0-\varepsilon, t_0+\varepsilon)\to\mathbb{R}^2$be a$C^{1}$application, with$\mathbf{r}'(t)=(0,0)\Longleftrightarrow\ t=t_0$, and $$\mathbf{r}(t)=\begin{cases} \mathbf{r}_1 (t)... 1answer 137 views ### Nowhere vanishing, normalized vector field with bounded derivatives It is well-known that any non-compact manifold admits a nowhere vanishing vector field. If we have a Riemannian metric we may pick such a vector field and normalize it so that at every point it has ... 1answer 84 views ### Cayley Subspaces in a Calibrated 8-Space Suppose we are given (\mathbb{R}^8,\Phi), where \Phi is the self-dual 4-form that defines Spin(7)\subset SO(8) (Cayley calibration, see Notes on the Octonians, page 23). Now some 4-subspaces V ... 1answer 127 views ### Uniqueness of a smooth function [closed] Let x,y:[a,b]\to\mathbb{R},\ a<b, a,b\in\mathbb{R} be two smooth functions (x,y\in C^{\infty}([a,b])). How can I prove that there is a unique function \theta:[a,b]\to\mathbb{R},\ \theta\in C^{\... 1answer 285 views ### is there a diffeomorphism with only finite orbits but of infinite order? I asked this in stackexchange, but got no answer, so I am trying here. Is it possible for a diffeomorphism \phi (of a smooth manifold M) to have the following properties: All its orbits are ... 1answer 265 views ### symplectic reduction for pair (M,D) Let M be a symplectic manifold with divisor D. Then how can we define symplectic reduction for pair (M,D)? 1answer 121 views ### Real algebraic surface Let f(u,v,z)\in \mathbb{Q}[u,v,z] be a polynomial in three variables such that X_{\mathbb{R}}\subset \mathbb{R}^{3} (the associated surface of real solution) is smooth. Suppose that the set of ... 3answers 611 views ### When a homeomorphism is a diffeomorphism w.r.t to a suitable smooth structure? Assume we have a homeomoprhism \phi:M\rightarrow M, where M is a topological manifold which admits at least one smooth structure. Is it always possible to construct a smooth structure on M w.r.... 0answers 58 views ### References to study Weak and Strong Topologies and aproximations on function spaces of manifolds I´m studing weak and strong topologies and aproximations on the function space C^{\infty}(M,N) of two manifolds M and N. I´m using the book Differential Topology of Morris Hirsch but it is a ... 2answers 421 views ### Charts needed for an atlas I just read this question link and asked myself, if there is any easy way to decide how many charts you actually need to cover a given compact manifold in \mathbb{R}^3, maybe at least in this ... 1answer 138 views ### Polygons with centroid at origin and vertices on compact codimension one submanifold of \mathbb{R}^{n}-\{0\} Let M be a compact codimension one submanifold of \mathbb{R}^{n} which does not contaion 0 and the origin lies in the bounded component of\mathbb{R}^{n}-\{0\}. Is it true to say that: ... 0answers 239 views ### manifold branched covering space for orbifolds An orbifold structure on some topological space X is a covering of X with local quotient charts V/G, where V is some connected manifold and G effectively acts on V via a finite group of ... 0answers 156 views ### (co)limits in the category of diffeological spaces vs. category of smooth manifolds I am wondering which (co)limits that exist in the category of smooth manifolds are preserved by the inclusion into the category of diffeological spaces? Are there any results that allow us to ... 0answers 80 views ### Reference request: linearly independent cycles in a manifold The following seems to be well known to experts, but I would be happy if there is a paper or textbook that I can cite. Note: all of the manifolds are assumed to be without boundary. Suppose that C ... 0answers 257 views ### Reference for a proof of the fiberwise Stokes theorem The fiberwise Stokes theorem says that given a differential form on a smooth fiber bundle whose fibers have boundary, the difference between the fiberwise integral of the differential and the ... 0answers 62 views ### Cohomology operators inducing local basis of 1-forms Suppose that \partial is a non-trivial (\partial \neq 0) cohomology operator on an m-dimensional manifold M (that is: \partial:\Omega(M)\to\Omega(M) is a degree 1 derivation such that \... 1answer 397 views ### Ehresmann fibration theorem for manifolds with boundary All manifolds in consideration may have nonempty boundary and may be disconnected. Let me fix a definition first. A map between smooth manifolds M\rightarrow N is a fiber bundle, iff it's locally ... 2answers 506 views ### Relating different topologies on C^{\infty}_c(M) This is somehow connected to this question. I can think of at least four topologies to put on C_c(M): Topologize C^{\infty}_c(M)\subseteq C^{\infty}(M) as a subspace with the weak Whitney C^\... 0answers 272 views ### Schoenflies and symplectic topology The final report from a workshop on Morse theory in low-dimensional and symplectic topology includes the following question, posed by Michael Hutchings: Can we apply symplectic geometry to solve the ... 1answer 130 views ### Are normal deformations of an embedding open in the C^{\infty}-space of embeddings of a compact smooth manifold` Let M be a compact smooth manifold (closed, for simplicity), n\in\mathbb{N} and equip the space of embeddings Emb(M,\mathbb{R^n}) wih the Whitney-C^{\infty}-topology. (The weak and the strong ... 1answer 235 views ### Strange problem about triplets of differential forms Suppose we have the following map:$$(\Omega^1(\mathbb{R}^n))^3\longrightarrow(\Omega^2(\mathbb{R}^n))^3(\alpha,\beta,\gamma)\longmapsto(\mathrm{d}\alpha+\beta\wedge\gamma,\mathrm{d}\beta+\... 2answers 832 views ### Stokes theorem with corners I've found the following version of Stokes' theorem in G. Stolzenberg's lecture notes 19: Notation: for$1 \le n \le m\Lambda(m, n) = \{ \lambda: \{1,...,n\} \rightarrow \{1,...,m\} \ | \ \...
To define characteristic classes in smooth vector bundles $E\longrightarrow M$ there is a more or less standard procedure: to choose a connection $\nabla$ and to derive the curvature $\Omega$, which ...