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].

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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 \...
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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 $\...
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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 ...
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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,...
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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/~...
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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 ...
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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-...
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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 ...
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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-...
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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?...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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,\...
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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 ...
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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 ...
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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 ...
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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 $U_i$ ...
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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 ...
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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 $M$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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+...
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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 (...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $S$....
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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 ...
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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 $...
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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 ...
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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 ...
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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 $...
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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 ...
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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 $...
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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 \...

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