# Tagged Questions

**0**

votes

**0**answers

40 views

### Total differential of Lipschitz submanifolds embedding

My interest is analysis on Lipschitz manifolds. I want to define traces of differential forms on a Lipschitz submanifold $N$ of a Lipschitz manifold $M$. In other words, I want to push-forward ...

**4**

votes

**2**answers

195 views

### Rank of a jet bundle of a vector bundle. Interpretation of the first jet bundle

I am trying to understand the jet bundles but currently I am stuck on the following questions:
Let $\pi: E\rightarrow X$ be a smooth (holomorphic) vector bundle of rank $k$ over a smooth (complex) ...

**4**

votes

**1**answer

170 views

### Exercises around Diffeological Spaces or a Diffeologic Atlas Theory

Is there a Book or a bunch of exercises to get used to diffeological spaces from a practical point of view? It seems to me that papers on this topic are mostly concerned with their very good ...

**10**

votes

**0**answers

150 views

### A simple proof that parallelizable oriented closed manifolds are oriented boundary?

So let $M$ be a smooth closed orientable real manifold such that $M$ is parallelizable, i.e., the tangent space $TM$ of $M$ is trivial. From the triviality of $TM$ we get that the Stiefel-Whitney and ...

**1**

vote

**0**answers

66 views

### Boundary of fibers of submersions

Let $M$ be a smooth manifold with boundary (say of dimension $m$) and let $N$ be a smooth manifold with no boundary (say of dimension $n$ with $m\geq n$). We have the following classical result:
...

**5**

votes

**2**answers

270 views

### A good metric for transversal intersections

Let $V_1,\ldots,V_k$ be a transversal set of smooth compact orientable sub-manifolds of a compact orientable manifold $M$, and set $V=\bigcap V_i$.
Is it always possible to equip a neighborhood $U$ ...

**-1**

votes

**2**answers

158 views

### Let be $f \in Diff(M)$. What we can say about the subgroup $span{f}< Diff(M)$? What are implications in the structure of $f$ and $M$? [closed]

Let be $f \in Diff(M)$. When is finite the subgroup $span\{f\}< Diff(M)$? What are implications in the structure of $f$ and $M$?

**2**

votes

**2**answers

150 views

### Topological invariants of toroidal orbifolds

Which are the most powerful topological invariants of toroidal orbifolds?
In particular I am looking for topological invariants of two-dimensional toroidal orbifolds such as $T^{2}/Z_{k}\times Z_{k}$ ...

**3**

votes

**1**answer

186 views

### Is any smooth homeomorphism isotopic to a smooth embedding?

Let $f:D^m\to \mathbb{R}^m$ be a smooth map ($D^m$ is the unit ball).
We call $f$ embedding if it is a homeomorphism on the image and the
derivative $D_xf$ is nonsingular at each point $x\in D^m$ ...

**3**

votes

**3**answers

290 views

### Finiteness of De Rham cohomology of smooth quasi-projective varieties

Let $U$ be a smooth quasi-projective variety over $\mathbf{C}$. Let $U^{\infty}$
be $U$ but thought of as a smooth manifold.
Q1: Is there a simple proof (so it should avoid Hironaka's ...

**1**

vote

**1**answer

133 views

### Normal tubular neighborhood theorem for semi(or pseudo)-riemannian manifolds

Suppose you have a manifold $M$ and a closed sub-manifold $A$, and let $g$ be a semi-riemannian metric,ie, $g_x$ defines a quadratic form on $T_xM$ such that $g_x(v,v)\ge0$, but $g_x(v,v)=0$ not ...

**2**

votes

**0**answers

49 views

### Embeddings of complex projective Stiefel manifolds

Let $k\leq n$ and define
\begin{align*}
V_k(\mathbb{C}^n) & =\{\textrm{orthogonal $k$-frames in } \mathbb{C}^n\}\\
& = \{A\in \textrm{Mat}_{n\times k}{(\mathbb{C})}\mid A^\ast A = {1}_k\}
...

**6**

votes

**1**answer

243 views

### What is the relation between Lefschetz fixed point theorem and Poincare-Hopf theorem on vector fields?

In Dubrovin/Fomenko/Novikov Modern geometry--Methods and applications, Part II, the (Poincare-)Hopf theorem is treated in section 15.2 (see theorem 15.2.7 on page 129), while the Lefschetz theorem on ...

**4**

votes

**0**answers

78 views

### are smooth homotopic open embedding of Hilbert manifolds smoothly isotopic?

A theorem of Chapman (Chapman, T. A. Homotopic homeomorphisms of infinite-dimensional manifolds. Compositio Math. 27 (1973), 135–140) states that every two open embeddings $f,g:M\to N$ of Hilbert ...

**2**

votes

**1**answer

77 views

### An example to show that when $P$ is a complex polarization the subbundle $P+ \bar P$ is not necessarly involutive

I start my question with some motivation. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if
$P$ is Lagrangian
P involutive
dim$P\cap\bar ...

**1**

vote

**0**answers

94 views

### How to show the Euler Characteristic is equal to self-intersection number of zero-section [duplicate]

myThe definition of the Euler characteristic (given in Guillemin and Pollack's "Differential Topology") of a compact oriented manifold $X$ is the self-intersection number of the diagonal $\Delta$ in ...

**1**

vote

**0**answers

110 views

### Smooth structures on quotient space

Suppose $G$ be a discreat group acting on $\mathbb R^n$ freely via two different actions $\rho_1$ and $\rho_2$. Suppose that $\mathbb R^n/\rho_1$ is homeomorphic to $\mathbb R^n/\rho_2$. However the ...

**-1**

votes

**4**answers

284 views

### An isomorphism on space of smooth sections

Let $M$ be a smooth complex manifold and $L$ be a complex line bundle over $M$. Let $\Gamma(M,L)$ be the space of smooth sections. Why $\Gamma(M,L)$ is it isomorphic to
$$A=\{f:L^{\times}\to ...

**23**

votes

**8**answers

2k views

### Are there some other notions of “curvature” which measure how space curves?

I am learning differential geometry and have a few questions on curvature. -- Background:
Gauss invented "Gauss curvature" to measure how surface curves.
Riemann gives an ingenious generalization ...

**1**

vote

**1**answer

213 views

### (n-1)-dimensional normal currents and Smirnov's paper

I don't know much about currents, but I saw a paper of Smirnov which seems relevant to a problem I am working on. In the very last paragraph of page 848 of the following paper
...

**1**

vote

**0**answers

82 views

### Remark in paper by Lima about $\epsilon$-neighborhood type function on a closed manifold

This is a closing remark in a paper of Elon Lima, Separation Theorem for Smooth Hypersurfaces.
He proves a well known lemma: Suppose $M\subset\mathbb{R}^m$ is a compact, orientable, smooth ...

**3**

votes

**0**answers

197 views

### Looking for a necessary and sufficient condition for the polarization $\mathbb{P}$ being positive

My question is about positivity of polarization in Geometric quantization theory. Let $\mathbb{P}$, be a complex polarization on symplectic manifold $(M,\Omega)$. For every $m\in M$, we can define a ...

**15**

votes

**2**answers

760 views

### Converse to Stokes' Theorem

Does satisfying Stokes' Theorem imply that a form is linear?
Let $M$ be an $n$-manifold. A differential $k$-form $\omega \in \Omega^k M$ assigns to each point $x \in M$ a function $\omega_x : ...

**3**

votes

**1**answer

211 views

### on Brieskorn Manifolds

Brieskorn showed that $X_{k}=\{(z_0, \dots, z_k)\in
\mathbb{C}^{k+1}| -z_{0}^{3}+\sum_{i=1}^{k} z_{i}^{2}=0\}$(k odd, k>2) is a
topological manifold. Is it a smooth manifold?
In general, let $a_1, ...

**0**

votes

**0**answers

166 views

### Fractional degree of a map?

Is there some natural notion of a fractional degree of a map?
The degree of a map is a generalization of the winding number,
and fractional winding numbers appear in the (mathematical physics)
...

**4**

votes

**2**answers

208 views

### good reference on brieskorn manifold

I am trying to learn something on the Brieskorn manifold (interested in the topological property)
Can the Mathoverflow Experts give me some good refencece (in English)?
By the way,is there an ...

**2**

votes

**0**answers

74 views

### Are Sasakian metrics (associated to bumpy metrics) bumpy?

Some background :
(1) Let $(M,g)$ be a smooth Riemannian manifold. Let $LM$ be the free loop space, the space of loops in $M$ of Sobolev class $W^{1,2}$. There is the energy functional $E:LM\to ...

**2**

votes

**1**answer

171 views

### Is it possible to approximate an area-preserving diffeomorphism by a sequence of conjugates of periodic rotations?

Is it possible to approximate an area-preserving diffeomorphism $T$ of the disk $\mathbb{D}^2$ by a sequence of conjugates of periodic rotations $B_n^{-1} S_{\frac{p_n}{q_n}} B_n$, where $ ...

**1**

vote

**0**answers

74 views

### Pseudo-Euclidean orbifolds

Are there any papers (reviews) devoted mainly to pseudo-Euclidean orbifolds in mathematics and physics (e.g. string theory)? A more specific question is related to orbifolds of type $\mathbb ...

**3**

votes

**1**answer

285 views

### Spin structures and divisibility of cohomology classes

Let me begin with some motivation. In calculating the Chern-Simons invariant of a $U(1)$ connection $A$ on a 3-manifold $M$, we can proceed by picking a bounding 4-manifold $X$ with $\partial X = M$ ...

**1**

vote

**2**answers

285 views

### even dimensional manifold homotopic to a symplectic or complex manifold

I have the following question: Let $M$ be an even dimensional Riemannian manifold. Under which conditions does there exists a homotopy to some symplectic manifold? is there any chance that such a ...

**1**

vote

**0**answers

96 views

### Extension of diffeomorphisms preserving bilateral bounds of the derivatives

Suppose $f$ is a $C^k (1\leq k\leq\infty)$ function from the unit ball $\mathcal{B}$ in $\mathbb{R}^n$ to itself, which is a diffeomorphism from the domain to its image, with the upper and lower ...

**1**

vote

**1**answer

162 views

### Understanding maps from M to R^n, for n>dim M

I am interested in "approximating" smooth maps from a compact smooth manifold $M$ of dim $m$ into $\mathbb R^n,$ for $n>m,$ by "nice" maps, with properties similar to those of Morse functions. Of ...

**2**

votes

**0**answers

156 views

### Stratification of a smooth map

So, this is an exercise. But from math.stackexchange I have been suggested to post this question here.
To find the Thom-Boardman stratification of the smooth map
...

**0**

votes

**1**answer

195 views

### Does Frobenius theorem apply to vector-valued function?

We know Frobenius theorem handle pde systems like
$\{Xf=0, Yf=0\}$
requiring Lie bracket $[X,Y]\equiv 0 \mod X, Y$ for completely integrability of the system. However, how to handle systems like ...

**2**

votes

**3**answers

220 views

### Fatou sets and topological entropy

Let us consider a diffeomorphism of a compact real manifold (complex manifold defined over the reals), and let us say that the diffeomorphism is birational. Hence, it extends to a birational map from ...

**6**

votes

**2**answers

302 views

### Exotic 4-manifolds with even positive partial betti number

It seems that usually smooth structures on compact 4-manifolds are distinguished by Seiberg-Witten/Donaldson invariants. And I don't know another direct way to do that. But in the case when $b_2^+$ is ...

**0**

votes

**0**answers

97 views

### transverse intersection of Whitney stratifications

Let $M$ be a smooth manifold. If $X$ and $Y$ are two Whitney objects, i.e. subsets with a given Whitney stratification, then $X$ and $Y$ are transverse if each stratum of $X$ is transverse to each ...

**0**

votes

**0**answers

289 views

### Does extending a section by the exponential map make it transverse to the zero set?

Let $V_1, V_2 \rightarrow M$ be two smooth vector bundles over
a smooth riemannian manifold $M$ and $s_1:M\rightarrow V_1$ a section
transverse to the zero set and $s_2: s_1^{-1}(0) \rightarrow V_2 ...

**1**

vote

**0**answers

80 views

### Problem reduced to analyzing solutions of a family of nonlinear systems of equations

I was able to reduce a research problem relating to normal numbers to analyzing the solutions of the following family of nonlinear systems of equations:
\begin{align*}
c_0+c_1+\cdots+c_{t-1}&=2t\\
...

**12**

votes

**2**answers

785 views

### Converse of Poincaré-Hopf theorem

Let $M$ be a connected, compact, oriented manifold of dimension $n<7$. If any two maps $M \to M$ having equal degrees are homotopic, must $M$ be diffeomorphic to the $n$-sphere?

**6**

votes

**1**answer

214 views

### vanishing of vector field in infinite dimensions

A simple fact: Given a vector field on a compact manifold with boundary, if the vector field points inward along the boundary, then it must vanish somewhere in the interior. (EDIT: As pointed out in ...

**4**

votes

**2**answers

261 views

### Is it impossible for the dimension of a topological space to increase under a smooth map?

First let me make a definition. Let $M$ be a smooth manifold and
$S \subset M $ a topological subspace of $M$. We say that $S$ has
"dimenion" at most $k$ if $S$ is a subset of
$$ X_1 \cup X_2 \ldots ...

**0**

votes

**1**answer

106 views

### A version of implicit function theorem when sections are not everywhere smooth?

Let $V_1, V_2 \rightarrow M $ be smooth vector bundles over a manifold $M$ and $s_1: M \rightarrow V_1$
a smooth section transverse to the zero set and $s_2: M \rightarrow V_2$ a continuous section
...

**1**

vote

**0**answers

116 views

### Can one always extend a smooth section defined on a non compact submanifold to the whole manifold, provided it extends continuously to the closure?

Let $V \rightarrow M$ be a smooth vector bundle over a smooth compact manifold $M$
(without boundary) and $X \subset M$ a smooth submanifold of $M$, that is not necessarily closed.
Suppose $s: X ...

**9**

votes

**0**answers

277 views

### 3 manifolds with diffeomorphic unit tangent bundles

What can one say about two closed oriented 3-manifolds $M_1$ and $M_2$ such that $S^2 \times M_1$ is diffeomorphic to $S^2 \times M_2$?

**1**

vote

**0**answers

240 views

### Tensors as multilinear maps

I am aware that many books on differential geometry define tensors as multilinear maps. Namely
$$
V\otimes W := L_2(V^* \times W^*,\Bbb F)
$$
I am also aware that this space is isomorphic to the ...

**5**

votes

**0**answers

223 views

### Quotient of 3-sphere by binary octahedral group?

Consider the Lie group $Spin(3)$, which can be thought of geometrically as the 3-sphere (e.g., it can be represented by the collection of unit quaternions). The quotient $Spin(3)/\pm I$ yields the ...

**5**

votes

**0**answers

210 views

### Existence of particular embeddings in euclidean spaces for non compact manifolds

Let $M$ be a $n$-dimensional smooth non-compact manifold such that the singular cohomology groups $H^{k}(M,\mathbb{Z})$ are finitely generated for $k\geq 0$. Can we find a sufficiently big integer ...

**10**

votes

**1**answer

442 views

### Strong Whitney embedding theorem for non-compact manifolds

$\newcommand{\RR}{\mathbb{R}}$The present question arises from some confusion on my part regarding the precise statement of the strong Whitney embedding theorem for non-compact manifolds.
The strong ...