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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9 votes
1 answer
494 views

Submersion vs fiber bundle

If one starts with a fiber bundle $f: X \to Y$ so that fibers having trivial integral homology by using spectral sequence one can get the induced map $f_*: H_*(X;\mathbb{Z}) \to H_*(Y;\mathbb{Z})$ is ...
52 votes
15 answers
21k 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 ...
2 votes
0 answers
347 views

$$ \left(\frac{\text{Man}^{\text{fr}}}{\text{Cobordism}},\coprod,\times \right)\simeq \left((\text{Fin}^{\simeq},\coprod)^{\text{gp}},\times\right)?$$ [closed]

If we combine a theorem of Pontryagin and the Barratt-Priddy-Quillen theorem we get that both sides of $$ \left(\frac{\mathrm{Man}^{\mathrm{fr}}}{\mathrm{Cobordism}},\coprod,\times \right)\simeq \left(...
2 votes
0 answers
38 views

Under what conditions principal directions define an integrable distribution?

Consider a hypersurface $M^n \subset \mathbb{R}^{n+1}$ which is compact without boundary. Assume that its second fundamental form $A$ has distinct eigenvalues $\lambda_1<\ldots<\lambda_k$ (with $...
1 vote
0 answers
64 views

Metric of negative curvature on connected sum

Let $(M_1,g_1)$ and $(M_2,g_2)$ be two Riemannian manifolds of dimension $n\geq 2$. If we consider the connected sum $M=M_1\mathbin{\#}M_2$ of the two manifolds; can one get a smooth metric $g$ on $M$ ...
1 vote
0 answers
113 views

Why does the bordism category have duals?

The proof of the Cobordism Hypothesis outlined by Lurie seems to assume that the $(\infty, n)$-category $\mathbf{Bord}_n^{(X, \zeta)}$ has duals, i.e. duals for objects and adjoints for all $k$-...
2 votes
1 answer
154 views

Order of a loop around a cone point

Let $M$ be a connected orientable two-dimensional orbifold with only cone points as singular points. Assume that $M$ has genus $\geq 1$. Let $\alpha$ be a loop around an order $p$ cone point. Can we ...
8 votes
2 answers
533 views

Presentations of exotic 4-manifolds

TLDR I want to see more examples of exotic $4$-manifold (hopefully connected, simply connected, oriented, and closed). Are there known presentations of $4$-manifolds $M$ with exotic structures, ...
4 votes
0 answers
287 views

Milnor’s smoothed corners technique for a product of manifolds with boundary and boundary defining functions

Let $M$ be a smooth manifold that we view as the interior of a compact manifold with boundary $\overline{M}$. Let $\rho$ be a boundary defining function for $\overline{M}$, i.e. $\rho$ is smooth, $\...
6 votes
1 answer
232 views

Why does Bott's obstruction theorem imply the vanishing of some cohomology classes of $B\Gamma_q$?

Recall that Bott's obstruction for integrability [Bott70] asserts that: Given a smooth (=$C^\infty$) $m$-manifold $M$ and a completely integrable vector subbundle $E\subset TM$ of rank $m-q$, every ...
2 votes
0 answers
47 views

How is the $k$-times iterative frame bundle $FF\cdots FM$ associated to the higher order frame bundle $F^k M$?

$\DeclareMathOperator\Gl{Gl}$As I understand it a natural bundle is one for which a diffeomorphism on the base space lifts to an automorphism on the total space of the bundle. It is my understanding ...
0 votes
0 answers
34 views

Estimate the gradient (with respect to local coordinates) of a partition of unity on a manifold

Suppose $\{U_\alpha\}$ is an atlas of coordinate patches of a (noncompact) smooth manifold $M$ of dimension $n$, with coordinates $(x_\alpha^1,\dots,x_\alpha^n)$ on $U_\alpha$. Furthermore we assume ...
0 votes
0 answers
123 views

Noether's formula for real algebraic surfaces

Is there a version of Noether's formula for the Euler characteristic of a surface for Real algebraic surfaces? Specifically, given $X$ a real algebraic compact smooth surface, what is the relationship ...
-4 votes
1 answer
301 views

Does a coarser topology lead to a non-Hausdorff topological manifold? [closed]

Take a topological manifold $M$. Suppose one considers a strictly coarser topology than the manifold topology. Can such topology result in a non-Hausdorff topological manifold? NOTE: PLEASE avoid the ...
9 votes
1 answer
322 views

Disintegration measures and differential forms

Let $X$ and $Y$ be smooth oriented manifolds of dimension $m$ and $n$, and let $f\colon X\to Y$ a proper smooth map. There is a theorem called the "Disintegration Theorem" which says ...
3 votes
1 answer
306 views

What is known about the "stickiness" of a smooth manifold to its tangent space?

Consider an $m$-dimensional smooth manifold $M$ embedded in $n$-dimensional real Euclidean space $\mathbb{R}^n$ ($n>m$). Consider a point $x \in M$ of the manifold, and for simplicity, choose the ...
-4 votes
1 answer
123 views

7-sphere x 4-sphere manifold and its physical significance [closed]

I am looking for sources about this manifold 7-sphere*4-sphere and its relations to theoretical physics. Where to go to read about 7-sphereX4-sphere manifold and its physical significance?
2 votes
0 answers
66 views

Assumptions for uniform measure of SDE on manifolds

Suppose we're working on a compact, Riemannian manifold $M$. Suppose $dX_t = -b(X_t, t)\,dt + \sigma^2 \,dB_t$ is started at the uniform measure on $M$. What kind of assumptions on $b$ make it so that ...
2 votes
0 answers
192 views

Smooth compactification of complex varieties and uniqueness

Since I'm working in differential geometry, for the following I'm strictly interested in the smooth setting over $\mathbb{C}$ and its relation to the setting over $\mathbb{R}$. Here are a few useful ...
4 votes
0 answers
159 views

Possible Euler characteristics of manifolds with tangential structures

Let $p:B\to BO$ be a fibration. We say that a manifold has a $B$-structure if its stable tangent bundle lifts to $B$. I am interested in the question of whether there exists, for a given even ...
5 votes
1 answer
216 views

Clarifying a result of Klingenberg

I am looking for some clarification on a result of Klingenberg. For context, I have a complete Riemannian manifold for which I would like to compute a lower bound on the injectivity radius at each ...
6 votes
0 answers
141 views

Extending topological vector bundles and obstruction theory

This is a question that has appeared in various forms on MathOverflow, see here and here, for example. But as opposed to these more algebraic questions, I am interested in the purely topological ...
3 votes
1 answer
164 views

Isomorphism between tangent bundle of $S^2$ and the kernel of a bundle homomorphism

Let $S^{4n+3} \to \mathbb{H}P^n$ be the standard projection which is a fiber bundle with fiber $S^3$. By the action of $S^1$ on $S^3$ we get a fiber bundle $$ \mathbb{C}P^1 \xrightarrow{\iota} \mathbb{...
1 vote
1 answer
192 views

Tightening a loop

Consider two $d$-dimensional convex polytopes $c_1, c_2$ that share a $(d-1)$-dimensional face $f$. Let $M$ be a surface ($2$-manifold) that intersects each of $c_1$ and $c_2$ in a $2$-ball. Suppose ...
4 votes
2 answers
530 views

Compactification of a product of manifolds

Let $M$ be a smooth manifold. We make the assumption that $M$ can be viewed as the interior of a compact manifold with boundary $\overline{M}$. In practice, for an explicit manifold, any ...
11 votes
2 answers
1k views

Thom's first isotopy lemma

Thom's first isotopy lemma says that given $f:M\to P$ a smooth map between smooth manifolds and a closed Whitney stratified subset $S$ of $M$, such that $f|_S:S\to P$ is proper and $f|_X:X\to P$ is a ...
0 votes
1 answer
108 views

How to estimate the distance between geodesics and points for Riemannian manifold with positive sectional curvature

Assume that $ M $ is a complete Riemannian manifold and there exists $ k>0 $ such that $ K(q)\geq k $ for any $ q\in M $, where $ K $ is the sectional curvature of $ M $. Let $ \gamma $ be a closed ...
1 vote
1 answer
174 views

Invariance of mutual information under injective functions

Let $X\colon \Omega\to\mathcal X$ and $Y\colon \Omega\to \mathcal Y$ be two random variables. In M.S. Pinkser's Information and information stability of random variables and processes mutual ...
0 votes
1 answer
101 views

Geodesic whose one end is at a ideal point

We know that every hyperbolic manifold $M$ is locally isometric to hyperbolic plane $\mathbb{H}_2$. Can we consider the inverse image, under the isometry, of the geodesic $\gamma$ that connects a ...
5 votes
1 answer
215 views

Commutative/ symmetric second covariant derivative

Consider a smooth manifold $M$ together with an affine connection (or covariant derivative) $\nabla$ on the tangent bundle $TM$. Is it possible to have an affine connection, possibly with non-zero ...
4 votes
0 answers
111 views

Finding inverses of certain elements in the set of normal invariants of a smooth manifold

Let, $V$ denote the Stiefel manifold of 2-frames $V_{10,2}$ . Consider the the map $S_\text{diff} (V) \xrightarrow{\eta} N_\text{diff} (V) $ in the surgery exact sequence of a smooth manifold. . ...
0 votes
0 answers
49 views

Symmetric indefinite matrix of fixed rank — manifold structure?

I have been studying symmetric indefinite matrices of fixed rank, which have been rather useful for a particular application. I wonder if there is a way to parameterise these by a smooth manifold, e.g....
3 votes
0 answers
182 views

An almost complex structure on $\Bbb S^n$ induces a cross product on $\Bbb R^{n+1}$

It is known that the only spheres that admit an almost complex structures are $\Bbb S^2$ and $\Bbb S^6$ (Borel and Serre, 1953). In particular, $\Bbb S^4$ cannot be given an almost complex structure (...
4 votes
0 answers
117 views

Triangulating piecewise-linear manifolds

Question 1: Is this the mainstream definition of a PL-manifold? Definition. A PL-manifold is a manifold with an atlas $(\varphi_i)_{i\in I}$ in which all transition maps $\varphi_j\circ\varphi_i^{-1}$ ...
3 votes
0 answers
185 views

Decomposition of forms on manifolds

Let $(M,g)$ be a globally-hyperbolic Lorentzian manifold, i.e. $M=I\times\Sigma$ with $I\subset\mathbb{R}$ being an open interval and $\Sigma$ a spacelike smooth Cauchy hypersurface. The metric is of ...
3 votes
0 answers
53 views

Jet at a singular point or a submanifold

Let $M$ be a smooth manifold, $p\in M$ and $f\in C^\infty(M\setminus\{p\})$. We will say that $f$ has a power-law singularity at $p$ of order $\eta$ if for every smooth immersion $\gamma:(-1,1)\to M$ ...
0 votes
1 answer
108 views

A sufficient condition for a collection of open sets of a manifold to contain all open sets

Question Let $k\geq 0$ be an integer and let $M$ be a topological $n$-manifold. Let $\mathcal{U}$ be a set of open sets of $M$ which satisfies the following closure properties: (1). Let $U\subset M$ ...
4 votes
1 answer
420 views

Detecting a "bad map" in Fintushel-Stern knot surgery

Background Let $X$ be a simply-connected smooth 4-manifold which contains a smoothly embedded torus $T$ with trivial normal bundle (in other words, $T^2\times D^2\subset X$). Let $K$ be a knot in $S^3$...
38 votes
4 answers
7k views

Two kinds of orientability/orientation for a differentiable manifold

Let $M$ be a differentiable manifold of dimension $n$. First I give two definitions of Orientability. The first definition should coincide with what is given in most differential topology text books, ...
5 votes
1 answer
328 views

To what extent differentiable mappings of an affine line into a manifold determine its differentiable structure? What about mappings of a plane?

If $M$ is a (real) differentiable manifold, its differentiable structure is completely determined if it is known which mappings $M\to\mathbf{R}$ are differentiable. How much can be said about the ...
7 votes
1 answer
699 views

Complex manifolds with corner?

I was reading Dominic Joycee article on Manifold with corner. He talk about manifold with corner modeled over $[0,\infty)^k\times \mathbb R^{n-k}$ for some $k\leq n$. From here I moved to Melrose ...
14 votes
2 answers
2k views

Is a manifold Euclidean if its tangent bundle is Euclidean?

I'm wondering whether an $n$-dimensional manifold diffeomorphic to $\mathbb{R}^n$ if its tangent bundle is diffeomorphic to $\mathbb{R}^{2n}$. Many thanks!
1 vote
1 answer
265 views

Cohomological dimension of kernel

Let $M$ and $N$ be two connected, smooth (possibly non-compact), aspherical manifolds (that is, $\pi_k(-)=1$ for $k\geq 2$) of dimensions $n$ and $n-r$ respectively. Let $f:M\to N$ be a smooth map ...
6 votes
1 answer
205 views

Existence of adjoint operators on manifolds

Let $(M,g)$ be an oriented Riemannian manifold and $V$ a finite-rank vector bundle equipped with a non-degenerate bundle metric $\langle\cdot,\cdot\rangle_{V}$. This bundle metric, in turn, gives rise ...
6 votes
0 answers
141 views

Kirby diagram of Enriques surface (as the "(1/2) K3 surface")

Not to be confused with $E(1)\cong\mathbb{C}P^2\#9\overline{\mathbb{C}P^2}$, which is also known as a $\frac{1}{2}K3$ surface (in the sense that removing a neighbourhood of a regular torus fiber in $E(...
5 votes
3 answers
204 views

First usage of the terms pseudo-isotopy and concordance in manifold theory

I am hoping I can use the collective knowledge of the forum to piece together some history. I'm wondering where the terms pseudo-isotopy and concordance originated, in their modern forms as used in ...
1 vote
0 answers
79 views

Is there a standard name for the following class of functions on non-Hausdorff manifolds?

Let $M$ be a (not necessarily Hausdorff) smooth manifold. Given an open chart $U\subset M$ and a compactly-supported smooth function $f:U\to\mathbb{R}$ on $U$, define $\widetilde{f}:M\to\mathbb{R}$ by ...
11 votes
2 answers
780 views

Neighborhoods of the identity in diffeomorphism groups

Finite dimensional Lie groups have the nice property that if $V$ is a small neighborhood of the identity, and $U \subset V$ another neighborhood, then $V$ is covered by $U^k$ (the set of all products ...
3 votes
2 answers
621 views

Simultaneous extensions of strongly convex functions

21/03/2017: I have decided to accept Denis Serre's answer, even though it does not exactly answer my question, however I like its simplicity and I'd say it is close enough to the desired claim. Of ...
3 votes
1 answer
325 views

When does a fibre bundle induce a long exact sequence in homotopy groups of mapping spaces?

Given a smooth fibre bundle of (compact, say) manifolds $F \to E \to M$, with $\pi : E \to M$ the projection, and $i_p : F \to E$ the inclusion of $F$ into any fibre $\pi^{-1}(p)$, we get, for any ...

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