**3**

votes

**0**answers

43 views

### Connected sum of chiral manifolds

Let $M,N$ be two closed, smooth, orientable manifolds of the same dimension and assume that these manifolds are chiral, i.e. they do not admit an orientation reversing automorphism. Then there are two ...

**1**

vote

**0**answers

35 views

### Measurability of eigenelements $s \mapsto (\varphi_k(s), \lambda_k(s))$ of Laplace-Beltrami on $M_s$

For each $s \in [a,b]$, let $M_s$ be a compact Riemannian manifold with no boundary.
Under what conditions on $s \mapsto M_s$ do the eigenvalues and eigenfunctions $(\varphi_k(s), \lambda_k(s))_{k ...

**1**

vote

**1**answer

126 views

### Sequence of smooth maps converging to the identity [on hold]

Let $\{g_{n}:B_{1}(0) \rightarrow \mathbb{R}^{2}\}_{n\in \mathbb{N}}$ be a sequence of smooth maps where $B_{1}(0)=\{x\in \mathbb{R}^{2} \mid |x|<1\}$ is the unit ball in $\mathbb{R}^{2}$. Assume ...

**4**

votes

**1**answer

181 views

### Are “Unions” of small exotic $\mathbb{R}^4$'s small?

Suppose $M$ is a smooth 4-manifold, and $U,V \subset M$ are exotic $\mathbb{R}^4$'s, i.e. homeomorphic to standard $\mathbb{R} ^4$.
Further more suppose $U$ an $V$ intersect nicely sucht that $U \cup ...

**3**

votes

**0**answers

75 views

### Intuitive Aproach of Dolbeault Cohomology

(Duplicated from math.stackexchange)
I would like to understand an intuitive approach to the definitions of Dolbeault Cohomology (using $\partial$ and $\bar{\partial}$) similar to the one given here. ...

**3**

votes

**2**answers

206 views

### Heat equation and evolution of number of critical points

Let $u_0$ be a smooth function on the unit sphere $S^1$ and assume that $u(t,x)$ is a smooth solution of the heat equation with initial data $u(0,x)=u_0(x)$. How one can apply the maximum principle to ...

**-1**

votes

**1**answer

103 views

### cartan killing metric [closed]

I know that we can define the killing form on a lie algebra. However, when going to the group manifold, does this give rise to a metric on the manifold? I thought that would be the case, but I cant ...

**0**

votes

**0**answers

86 views

### Functional Calculus and Fredholm index

Let $-\Delta: W^{2,2} \subset L^2(\mathbb{S}^2) \rightarrow L^2(\mathbb{S}^2)$. Then it is "easy" to show that $-\Delta $ is self-adjoint. Now, I am looking for closed operators $T$ and $T^*$ of order ...

**1**

vote

**1**answer

68 views

### Existence of a fixed-point free map in a manifold [closed]

I'm having some to proof a question. I have to show that a compact manifold that admits a nowhere vanishing smooth vector field has a smooth map fixed-point free homotopic to the identity map.
I know ...

**4**

votes

**1**answer

412 views

### Elementary Proof of the Uniqueness of Smooth Structures on R

Is there any 'elementary' proof of the uniqueness of smooth structures on $\mathbb{R}$? By elementary, I mean that the proof does not use any sophisticated topological machinery. In particular, I'm ...

**4**

votes

**0**answers

257 views

### A vector space associated with a vector field on a symplectic manifold

Let $(M,\omega)$ be a $2n$ dimensional symplectic manifold and $X$ is a smooth vector field on $M$. Consider the following subvector space of $\chi^{\infty}(M)$: $$S(X)=\{Y\in ...

**0**

votes

**1**answer

148 views

### Some quantities which definitions are (somehow) similar to the classical Divergence

Motivated by classical formulas $L_{X}=d\circ i_{X}+i_{X}\circ d$ and $L_{X} \Omega=Div(X) \Omega$ and the essential role of the diff operator $d$ in definition of divergence, we define some ...

**3**

votes

**1**answer

132 views

### Co-rank of a group with $a^2b^2c^2=1$ (fundamental group of non-orientable surface)

What is the co-rank of a group $$G=\langle a_1,a_2,\dots,a_h\mid a_1^2a_2^2\dots a_h^2=1\rangle,$$ that is, finitely generated group with $h$ generators and one relation?
By co-rank, I mean the ...

**24**

votes

**2**answers

1k views

### Unifying Geometry for Characteristic Classes

When working with characteristic classes (more concretely Chern classes), one finds at least four essentially distinct approaches:
Axiomatic Approach. See, for instance, Vector Bundles and K-Theory, ...

**1**

vote

**1**answer

183 views

### Futaki invariant on $X=Bl_p(\mathbb CP^2)$ for different line bundles

Let $X$ be a projective variaty which blow up at a point $p$ , i.e, $X=Bl_p(\mathbb CP^2)$, then for the Line bundle $L=-K_X$, we have for Futaki invariant $Fut_L\neq 0$, I want to see, what about ...

**0**

votes

**0**answers

98 views

### A consequenc of a Lie group act on a Riemannian manifold by isometry

I am learning differential geometry for using this topic in my research. I am stuck to prove following Result which I got in a article.
Formulation:
Let $ f: [0, 1]\rightarrow \mathbb{R}^2$ be a ...

**4**

votes

**1**answer

304 views

### Is the space of real conics with a singular point an orientable manifold?

Consider the space of non zero real homogeneous degree $2$ polynomials in three variables upto scaling. This space is $\mathbb{R} \mathbb{P}^5$. The zero set
of such a polynomial gives a real curve ...

**4**

votes

**1**answer

101 views

### New setting for tensor definition

This is the standard definition for a tensor in a smooth manifold (Nakahara, for example):
But while reading Salamon's Riemannian Geometry and Holonomy Groups I have found this rather different ...

**3**

votes

**1**answer

258 views

### Question about the h-principle

So generally we define a differential relation to be $\mathcal{R} \subset X^{(r)}.$ In the case that $X=M\times N$ is it possible to have $\mathcal{R}=X^{(1)}$? So in this case the formal solutions ...

**19**

votes

**0**answers

256 views

### Can one properly embed a differential manifold into numerical space of double dimension? [duplicate]

If $X$ is a $ C^\infty$ differential manifold of dimension $n$, then there exists an embedding $f:X\to \mathbb R^{2n+1}$.
This is a not too difficult theorem due to Whitney, proved in many textbooks.
...

**0**

votes

**1**answer

80 views

### Almost fixed point property

Let $X$ be a Hausdorff topological space with the following property:
For every continuous function $f:X\to X$, there is a finite subset $S\neq \emptyset$ of $X$ with $F(S)\subset S$
...

**2**

votes

**1**answer

154 views

### Divergence invariant lifting of a vector field via a submersion

What is an example of a smooth submersion $P:S^{3}\to S^{2}$ for which the following statment is Not true:
For every vector field $X$ on $S^{2}$ there is a non vanishing vector field ...

**0**

votes

**0**answers

77 views

### A question on tangent bundle (and second tangent bundle)

Let $M$ be a $n$ dimensional manifold and $p:TM\to M$ be the projection map. Then $\ker Dp$ is a $n$ dimensional vector bundle on $TM$, as a sub bundle of $TT(M)$.
For what type of manifolds, ...

**5**

votes

**1**answer

167 views

### Is the unit tangent bundle of $S^{n}$ parallelizable?

Is the unit tangent bundle of $S^{n}$ a parallelizable manifold. This is motivated by the fact that $TS^{n}$ is parallelizable?

**4**

votes

**2**answers

123 views

### functions which covers(good covers) manifolds

Let $M$ be a (not necessarily compact)) smooth manifold.
1.Is there a smooth map $f:M\to \mathbb{R}$ and an open covering $\mathbb{R}=\cup U_{\alpha}$ such that each $f^{-1}(U_{\alpha})$ ...

**8**

votes

**1**answer

273 views

### Existence of certain “nondegenerate” function and manifold topology

Let $M$ be a smooth manifold without boundary, not necessarily compact.
Let $f$ be a real-valued smooth function on $M\times M$. We say $f$ is good if for any point $(x,y)\in M\times M$ with local ...

**1**

vote

**1**answer

154 views

### Free action of symmetric groups

What type of compact manifolds, can be acted freely by symmetric group $S_{m}$ for some $m>2$?
Is there a compact manifold which can be act freely by all symmetric ...

**2**

votes

**2**answers

206 views

### A metric associated with a continuous surjective map $f:X\to Y$

Assume that $f:(X,d_{1})\to (Y,d_{2})$ is a continuous surjective map between compact metric spaces. We define another
metric $d_{f}$ on $Y$ With $$ d_{f}(y_{1},y_{2})=Hd(f^{-1}(y_{1}), ...

**14**

votes

**4**answers

418 views

### The limit of edge-midpoint convex polyhedra

Starting with a convex polyhedron $P_1 \subset \mathbb{R}^3$,
replace that with $P_2$, the convex hull of the midpoints of the edges of $P_1$.
Continuing this process, we obtain a ...

**6**

votes

**2**answers

219 views

### Homeo-Fixed point property

Edit: According to comment of Michał Kukieła I revised the question
A topological space $X$ satisfies "Homeo-fixed point" property if every homeomorphism $f$ on $X$ possess a fixed point.
...

**4**

votes

**0**answers

168 views

### $n$-Fold Framed Functions

Suppose that $M$ is a manifold. One can consider a suitably constructed space of generalized framed Morse functions on $M$, let's call it $\mathrm{Fun}^\mathrm{fr}(M)$. This space is known to be ...

**1**

vote

**0**answers

84 views

### Tubular neighborhoods in the proof of the Morse homology theorem

I have a question regarding the proof of the Morse homology theorem given by D. Salamon in "Morse theory, the Conley index and Floer homology". The full text can be found here:
...

**0**

votes

**2**answers

72 views

### A full dim. subvector space of $\chi^{\infty}(M)$ which all non zero elements are nonvanishing vec.field

What is an example of a $n$ dimensional manifold $M$ which is not a lie group or $S^{7}$ but satisfies the following property?:
There is an $n$ dimensional sub vector space ...

**0**

votes

**1**answer

147 views

### Can a smooth function on a cross be extended to the whole plane?

Consider a real function on the union of two lines R×0 and 0×R in R² whose restrictions to R×0 and 0×R are smooth functions R→R.
Is it possible to extend this function to a smooth function on R²?
...

**3**

votes

**0**answers

215 views

### Quotes from Connes

I found the following remark by Connes HERE:
"the main quality of the homotopy type of a manifold is to satisfy Poincare duality not only in ordinary homology but in K-homology with the Fredholm ...

**1**

vote

**0**answers

58 views

### Cover a set with balls centered at smooth functions (Ascoli theorem)

Assume $M$ to be a compact $n$-dimensional manifold, endowed with a complete metric.
Let us consider the space $C^\infty(M)$ endowed with the standard $C^\infty$ topology, i.e. generated by the ...

**4**

votes

**0**answers

182 views

### Atlas of a manifold as a Sheaf

--Hopefully this question does not dublicate another--
In this question Tom Goodwillie pointed out, that the 'atlas part' of
the definition of a smooth manifold can be redefined in terms of
sheaves. ...

**0**

votes

**0**answers

99 views

### A special Lie subalgebra

Motivated by comments of the following post A question on involutions on the Lie algebra of vector fields we ask the following question:
Let $L$ be a Lie algebra. We consider the Lie subalgebra ...

**12**

votes

**2**answers

543 views

### Does every compact manifold exhibit an almost global chart

Let $M$ be a compact connected manifold.
Is there a chart $\Psi:U \to \mathbb{R}^n$ such that the closure of $U$ is $M$?
This is true for $S^n, T^n, K$, all compact surfaces, etc.
If it is not true in ...

**3**

votes

**0**answers

102 views

### Is a continuous map between smoothable manifolds of the same dimension always smoothable?

(My question is inspired by this math.SE question, whose
negative answer I showed by a dimension-increasing map.)
Is it the case that for all smoothable manifolds $M_0$ and $M_1$ with the same ...

**0**

votes

**1**answer

65 views

### A question on parallelizability

Is there a manifold $M$ such that for every $x\in M$, $M-\{x\}$ is not parallelizable but there is a finite set $S\subset M$, with $\# S>1$, such that $M-S$ is parallelizable?

**2**

votes

**0**answers

149 views

### The Cauchy Problem in General Relativity: Existence of a Hausdorff Development

This is related to a problem that I posed about a year ago. I was given several references by a number of experts who were kind enough to entertain my rather arcane question. Those references were ...

**4**

votes

**1**answer

219 views

### The embeddings $S^{6} \to S^{7}$ and $S^{7}\to S^{8}$

Is the standard smooth structure of $S^{7}$ the only structure for which each of the following canonical embedding is an smooth embedding?:
1)$S^{6}\to S^{7}$
2)$S^{7}\to S^{8}$

**9**

votes

**1**answer

218 views

### heat kernel on n-sphere

I'm interested in diffusion, a.k.a. the heat kernel driven by the Laplace-Beltrami operator, on the $n$-dimensional sphere. There are lots of bounds showing that, for small times, it behaves in a way ...

**1**

vote

**2**answers

205 views

### A question on involutions on the Lie algebra of vector fields

Edite According to the essential comment of Ian Agol I revise the question as follows
For a smooth manifold $M$, is there a non identity involution $\theta$ on the lie algebra $\chi^{\infty}(M)$ ...

**4**

votes

**2**answers

284 views

### A question about Dehn surgery and Brieskorn homology 3-spheres

I have been learning about Brieskorn homology 3-spheres $\Sigma(a_1,...,a_n)$ and Seifert manifolds. My reference is the first few pages of Saveliev's "Invariants of Homology 3-spheres."
If I ...

**5**

votes

**2**answers

228 views

### What 3-manifolds can be obtained by gluing $ S^1 \times P $ and two copies of $S^1 \times D^2$

Let P denote the pair of pants e.g. a sphere minus three small discs $D_1$,$D_2$,$D_3$ about marked points $x_1,x_2,x_3$. I then consider $P \times S^1$. We have boundary components $T_1$,$T_2$,$T_3$. ...

**0**

votes

**0**answers

96 views

### Question about a particular estimate in Riemannian geometry

I have been studying the book Some Nonlinear Problems In Riemannian Geometry - Thierry Aubin. On page $46$ he begins the proof of the Sobolev imbedding theorem to manifolds. The proof is divided in ...

**23**

votes

**2**answers

681 views

### Ellipses on spheres (and other surfaces)

Define an ellipse $E$ on a sphere as the locus of points whose sum of
shortest geodesic distances to two foci $p_1$ and $p_2$ is a constant $d$.
There are conditions on $\{ p_1, p_2, d \}$ for this ...

**27**

votes

**9**answers

4k views

### Why differential forms are important?

Importance of differential forms is obvious to any geometer and some analysts dealing with manifolds, partly because so many results in modern geometry and related areas cannot even be formulated ...