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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2
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1answer
102 views

Sequence of smooth maps converging to the identity

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 ...
1
vote
0answers
24 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 ...
3
votes
2answers
199 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 ...
4
votes
1answer
159 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
0answers
68 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. ...
0
votes
0answers
81 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
votes
1answer
91 views

cartan killing metric [on hold]

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 ...
4
votes
0answers
256 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 ...
1
vote
1answer
67 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
1answer
405 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 ...
24
votes
2answers
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, ...
8
votes
1answer
341 views

Homotopy spheres with vanishing and non-vanishing $\alpha$-invariant

I'm unsure whether this question is appropriate for mathoverflow, so feel free to criticize. All manifolds are closed, smooth and have dimensions $n\ge 5$. The Atiyah-Shapiro-Bott-Orientation gives ...
0
votes
1answer
131 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
1answer
131 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 ...
-1
votes
0answers
87 views

Fundamental group of connected sum for non-orientable manifolds [migrated]

For orientable manifolds, $\pi_1(M\#N)=\pi_1(M)*\pi_1(N)$, fundamental group of connected sum is free product of fundamental groups. As far as I understand, for non-orientable manifolds connected sum ...
1
vote
1answer
180 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
0answers
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
1answer
303 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
1answer
100 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
1answer
255 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
0answers
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. ...
4
votes
0answers
167 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 ...
2
votes
1answer
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
1answer
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 $F(S)\subset S$ ...
0
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0answers
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, ...
6
votes
1answer
852 views

Does a Trivial Tangent Bundle Induce a Multiplication?

Let $M$ be a connected smooth manifold, and assume that it is parallelisable; that is, its tangent bundle is trivial. Does $M$ admit an H space structure? That is, does there exist a smooth map ...
5
votes
1answer
166 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
2answers
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
1answer
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 ...
2
votes
2answers
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}), ...
1
vote
1answer
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 ...
14
votes
4answers
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
2answers
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. ...
1
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0answers
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: ...
23
votes
2answers
678 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 ...
0
votes
2answers
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
1answer
145 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? ...
2
votes
1answer
199 views

Frobenius rank of a manifold

The rank of an smooth manifold M is defined by Milnor, as follows: "The maximum number of independent commuting vector fields on M" For example it is well known that the rank of $S^{3}$ is 1 (Lima, ...
26
votes
9answers
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 ...
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15answers
8k views

What is the Implicit Function Theorem good for?

What are some applications of the Implicit Function Theorem in calculus? The only applications I can think of are: the result that the solution space of a non-degenerate system of equations ...
3
votes
0answers
214 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
0answers
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 ...
9
votes
5answers
2k views

Are there two non-homotopy equivalent spaces with equal homotopy groups?

Could someone show an example of two spaces $X$ and $Y$ which are not of the same homotopy type, but nevertheless $\pi_q(X)=\pi_q(Y)$ for every $q$? Is there an example in the CW complex or smooth ...
15
votes
1answer
752 views

Periodic Orbit property

A topological space $X$ satisfies "Periodic orbit property", briefly POP, if for every continuous map $f:X \to X$, there exist a natural number $n$ and a point $x_{0}\in X$ such that ...
0
votes
0answers
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 ...
4
votes
0answers
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. ...
12
votes
2answers
541 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 ...
9
votes
1answer
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
2answers
204 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 $\chi^{\infty}(M)$ ...
3
votes
0answers
101 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 ...