Questions tagged [smooth-manifolds]
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].
970
questions
1
vote
0answers
28 views
domain of flow of an inner vector field is a manifold with corners
Crosspost.
Let $X$ be a manifold with corners. Let $\vec v$ be an inner vector field on $X$. The existence and uniqueness theorem for ODE says there's a domain of flow $\mathfrak D(\vec v)\subset \...
7
votes
1answer
170 views
How special are homogeneous spaces?
Let $M$ be a smooth finite dimensional manifold, how restrictive is it to require $M$ to admit a smooth action by a finite dimensional Lie group $G$?
Related questions/approaches: Of course we need $\...
1
vote
0answers
80 views
Can two smooth submanifolds intersect transversally on an almost Cantor set?
Note: In the following, all submanifolds are assumed to be smoothly embedded, i.e. the image of a smooth embedding.
Definition: A Cantor subset of a topological space $X$ is a closed, nowhere dense ...
1
vote
1answer
67 views
The tangent map of the multiplication map of a vector bundle
If $\beta: \mathbf{U}\times \mathbf{V}\to \mathbf{W}$ is a bilinear map between real linear spaces then its derivative at a point $(u,v)$ is given by the Leibniz rule $$D\beta(u,v)(u_0,v_0)=\beta(u,...
10
votes
0answers
420 views
Exotic analytic triangulations of $S^5$?
I would like to understand better the nature of bad triangulations of $S^5$, discussed in two Lectures of Jacob Lurie
https://www.math.ias.edu/~lurie/937notes/937Lecture2.pdf
http://www-math.mit.edu/~...
2
votes
0answers
127 views
Extending an embedding with trivial normal bundle
I am recently studying the book Notes on Cobordism Theory by R. E. Stong and I have noticed that the proposition below is (implicitly) used (for example to extend a $(B,f)$ structure on a boundary of ...
19
votes
1answer
735 views
Can the product of a 3-dimensional lens space with a circle be diffeomorphic to another such product when the lens spaces aren't diffeomorphic?
This is a question that I need to answer in order to resolve an issue for my dissertation and I am looking for a reference. Here is the precise statement of the question.
Suppose we have two three-...
2
votes
1answer
56 views
Positive scalar curvature on the double of a manifold (assuming mean convexity of the boundary)
This question is related to a previous one.
Let $(M^n,g)$ be a compact Riemannian manifold with boundary. Assume it has positive scalar curvature and $\partial M$ is mean convex (positive mean ...
2
votes
0answers
94 views
Diffeomorphism classification of manifolds with fundamental group $\mathbb{Z}_n$, $n>2$
I am looking for diffeomorphism classifications of manifolds with $\pi_1=\mathbb{Z}_n$, $n>2$. I only know of Ottenburger's PhD thesis, A diffeomorphism classification of 5- and 7-dimensional non-...
3
votes
1answer
85 views
Positive scalar curvature on the double of a manifold
Let $(M,g)$ be a compact Riemannian manifold with boundary and assume it has positive scalar curvature.
Question. Is it true that $DM$, the double of $M$, admits a metric of positive scalar curvature?...
1
vote
0answers
61 views
Uniqueness of collar neighborhoods for non-compact boundary case in smooth setting
Let $M$ be a smooth manifold and let $f_0, f_1 \colon [0, 1] \times
\partial M \to M$ be two smooth embeddings that are the identity map
on $\partial M \times\{0\} = \partial M$ . If $\partial M$ is
...
15
votes
1answer
481 views
Is there a $4$-manifold which Immerses in $\mathbb{R}^6$ but doesn't Embed in $\mathbb{R}^7$?
I'm interested in both version of the question in the title, i.e. in the topological category and in the smooth category. By a topological immersion I mean a local embedding. I was asking in ...
8
votes
1answer
277 views
Non orientable, closed manifold covered by two contractible charts
This is a follow up of my previous MO question "Non orientable, closed manifold covered by two simply-connected charts." Nick L's nice answer shows that such manifolds actually exist, ...
0
votes
1answer
93 views
Intersection Grassmanian planes
I am reading a paper that used Grassmanian planes properties. In particular, they studied the intersection of Grassmanian planes; they check the intersection Grassmanian of $n-k$-planes and ...
8
votes
1answer
524 views
Non orientable, closed manifold covered by two simply-connected charts
This question arose during my Differential Geometry course. Possibly there is an obvious answer, but I do not see it, and I could not find it in the literature. The same question was asked yesterday ...
0
votes
0answers
28 views
Preserving the Holomorphicity of a Complex Differentiable Form on a Polytope
I had originally intended the following to be a secondary question to my original post but then realized that it merited a separate entry entirely.
Question: Could it be possible to approximate a ...
4
votes
0answers
112 views
Sheaf-like reconstruction of a continuous function
Let $X$ and $Y$ be topological manifolds and let $\{(\phi_x,U_x)\}_{x \in X}$ and $\{(\psi_y,Y_y)\}_{y \in Y}$ be respective atlases of $X$ and $Y$; with each $\phi_x:U_x\rightarrow \mathbb{R}^n,\...
8
votes
1answer
254 views
Outer automorphism group of Brieskorn homology sphere?
In this post, it is discussed how a Brieskorn homology sphere $\Sigma(a_1,a_2,a_3)$ with $\displaystyle \frac{1}{a_1}+ \frac{1}{a_2}+ \frac{1}{a_3} < 1$ is an aspherical manifold with a ...
16
votes
0answers
489 views
Bernoulli & Betti numbers (of manifolds) and the prime 34511
The purpose of this question is to resolve a mystery surrounding the prime 34511 that has got me bogged down for a while now. If you only care about the number theory and not the motivation coming ...
1
vote
0answers
34 views
Vertical bundles of higher order tangent bundles
Let $M$ be a smooth (finite dimensional, Hausdorff and second countable) manifold. Let $T^kM$ be the manifold of equivalence class of curves that their derivates (in charts) agree up to order $k$. Let ...
-4
votes
2answers
258 views
Constructing a new manifold with a germ of manifold [closed]
Given a germ of manifolds and compatible Riemannian metrics, can we construct a new Hausdorff manifold using the exponential map?
A germ of manifolds at a point $m$ is a series of manifolds $U_i$ ...
4
votes
0answers
104 views
Involutions on $D^4$ with a fixed arc
By a theorem of Livesay, the 3-sphere has a unique (up to equivariant diffeomorphism) smooth involution with exactly two fixed points. Thinking of $S^3$ as the unit sphere in $\mathbb{R}^4$, this ...
2
votes
0answers
82 views
Rational systole of a manifold
I also posted this question on MSE, but since it may be a delicate question, I decided to post it here.
Given a Riemannian manifold $(M^n,g)$ and an integer $1 \leq k \leq n-1$, the $k$-systole of $M$ ...
2
votes
0answers
76 views
Canonical class & ring of projective space $\mathbb{P}^n$ in differential geometry
David Mumford remarks in his book Algebraic Geometry I, Complex Projective Varieties on
page 109 that the fact that the canonical ring $\oplus_{k=0}^{\infty} \Omega_{k, \mathbb{P}^n}$
of projective ...
1
vote
0answers
68 views
Codimension one submanifold gives cofibrant pair
Let $M$ be a smooth manifold, and $N$ be an embedded smooth submanifold of $M$ with $\partial M=\varnothing=\partial N$. Suppose, $\dim M-\dim N=1$, and $N$ is a closed subset of $M$.
Does the ...
6
votes
1answer
106 views
Tangential harmonic $1$-forms are pullbacks of harmonic functions
This question has also been posted on MSE, but maybe here is the right place to obtain an answer.
Let $(M^3,g)$ be a compact connected oriented Riemannian $3$-manifold with nonempty boundary. The ...
3
votes
1answer
169 views
Ricci flow for manifold learning
I know that mean curvature and diffusion-type flows are common in manifold learning because of their smoothing effects. I haven't seen Ricci flow used as much. Given that Ricci and diffusion-type ...
1
vote
0answers
52 views
When is the unstable direction map $x\mapsto e^{u}(x)$ injective?
Let $f:M \to M$ be a $C^{2}$-Anosov diffeomorphism. Therefore, there exists an invariant splitting of the tangent bundle $T_{x}M = E^s(x) \oplus E^u(x)$ into a stable and an unstable directions, that ...
12
votes
1answer
541 views
What is known about exotic spheres up to stable diffeomorphism?
In even dimensions $n=2k$ we can define two smooth manifolds $M$ and $N$ to be stably diffeomorphic if they become diffeomorphic after the connect sum with $r$ many copies of $S^k \times S^k$ for some ...
8
votes
0answers
107 views
singular homology of manifold with corners
Given two smooth manifolds with corners, let's say that a map $f:X\to Y$ is "transversally smooth" if it is smooth in the usual sense and if (in a local sense on $X$) for every open Whitney ...
4
votes
2answers
233 views
Is there any Lie groupoid structure on $Hom(\mathcal{G}, \mathcal{H})$ where $\mathcal{G}$ and $\mathcal{H}$ are Lie groupoids?
We know that in general, there is no smooth manifold structure on $Hom(X,\, Y)$ where $X$ and $Y$ are smooth manifolds, but under certain nice conditions (see https://ncatlab.org/nlab/show/manifold+...
6
votes
0answers
256 views
How much differs the category of real-analytic manifolds from $C^\infty$ ones?
I was thinking about the difference between the concept of real-analytic function (for any point the Taylor-series of $f$ converge to the function in a neighborhood of the point) and complex analytic (...
11
votes
0answers
224 views
Almost isometric manifolds are diffeomorphic
I am looking for a reference to the following statement.
(It should be known --- I saw it before, don't remember where; search by keywords did not help.)
Let $f\colon M\to N$ be a homeomorphism ...
1
vote
0answers
65 views
What 'large' surfaces are there?
I answered this question on "is there a longest geodesic" by a kind of a joke, which I couldn't resist: the long line! Simply going by the name it had to be the 'longest geodesic'! I didn't ...
2
votes
1answer
106 views
Approximating continuous functions via diffeomorphisms on compact manifolds
Let $M$ be a compact and connected manifold without boundary.
My question is how to prove the following fact which I believe is true:
If $f : M \to \mathbb{R}$ is a continuous function that attains ...
2
votes
0answers
81 views
Intuition behind Nakano positivity
I am learning about Nakano positivity of hermitian vector bundles, which is the strongest notion of positivity we can ask. I don't understand what is the geometric meaning of it. Let me briefly ...
4
votes
1answer
182 views
Path integral presentation of solutions of Dirac equation
It is well known how to present solutions on the heat equation using the path integral (including the case of Riemannian manifold).
Is there a way to present solutions of the Dirac equation using path ...
12
votes
0answers
161 views
Smooth dual cell structure
Let us consider a closed oriented smooth manifold M. It is well known that a smooth combinatorial triangulation can be constructed for it. That is to say, a homeomorphism from the geometric ...
2
votes
0answers
117 views
Equality of Hausdorff measure and Lebesgue measure on manifolds (reference)
Let $\mathcal{M} \subset \mathbb{R}^N$ be an $n$-dimensional $C^1$ submanifold (connected). We have two metric functions on $\mathcal{M}$:
The Euclidean distance inherited from $\mathbb{R}^N$.
The ...
3
votes
0answers
70 views
Restriction of complete 1-forms to closed submanifolds (Sharpe's book on Cartan geometries)
In his Book Differential Geometry: Cartan's generalization of Klein's Erlangen Program, Sharpe gives the following definition of a complete 1-form:
Soon thereafter he gives the following example:
I ...
4
votes
1answer
302 views
The maximum number of vertical independent vector fields on the tangent bundle
Let $M$ be a differentiable manifold.
Is there a name for the maximum number of globally defined independent vector fields on $TM$ which are tangent to the fibers of $TM\to M$? Is there a name for ...
1
vote
1answer
89 views
Realizing a set as the image of a smooth map
Consider the following subset of $\mathbb{R}^2$:
$S = \{ (x, y) \in \mathbb{R}^2 : |y| \leq |x|^{3/2} \}$
(See here for a plot on Wolfram Alpha.)
The origin $(0, 0)$ is a kind of singular point of $S$....
4
votes
0answers
90 views
Push forward of Chern character and index theorem
I have some trouble understanding a proposition in Leung's paper "Symplectic Structures on Gauge Theory" published in Commun. Math. Phys. 193, 47 – 67 (1998).
I expose here the setup for my ...
0
votes
0answers
126 views
Homology of a closed $3$-manifold with balls removed
This question has been posted on MSE with no answers.
Let $M^3$ be a closed, connected and orientable smooth $3$-manifold and let $\mathring{M}$ denote the manifold $M$ with $n$ disjoint open balls $...
1
vote
1answer
120 views
A question regarding the action of a Lie subgroup
Suppose $H$ is a closed subgroup of a Lie group $G$. Then in Lee's book Introduction to Smooth Manifolds (Ch. 9) he showed that the action $H\times G\to G$ $(h,g)\mapsto gh$ is a smooth, free, proper ...
0
votes
0answers
35 views
Symplectic form on the space of unitary connections $\mathcal{A}(E)$
Let $E\rightarrow X$ be a Hermitian vector bundle over a (Kahler) manifold $X$. The space of unitary connection $\mathcal{A}(E)$ is an affine space modelled over $\Omega^1(X,u(E))$ and is endowed with ...
1
vote
0answers
137 views
Number of smooth structures on a compact manifolds
I'm new to this field: is there a compact smooth manifold of dimension $n\geq 5$ with uncountably many smooth structures no two of which are diffeomorphic. This is motivated by Milnor's exotic sphere $...
-1
votes
1answer
91 views
Local triviality condition in vector bundles [closed]
Let $E$ and $M$ be smooth manifolds (of finite dimension, Hausdorff and second countable). Let $\pi:E\longrightarrow M$ be a surjective submersion such that:
$E_p:=\pi^{-1}(p)$ is a real vector space ...
2
votes
0answers
63 views
Geometric meaning of second tangent bundle, or of microsquares in SDG
In synthetic differential geometry (SDG), we have $D\subset R$ comprised of the second order nilpotents. The Kock-Lawvere axiom (KL axiom) implies that a function $D\times D\to R^n$ is of the form $...
4
votes
0answers
64 views
Fréchet subdifferentiation on riemannian manifolds
Context. I'm looking for a "natural" definition of subdifferentials on riemannian manifolds.
Given a function $F:\mathbb R^m \to \mathbb R$, its Fréchet-subdifferential at a point $w \in \...
