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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123
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How do I make the conceptual transition from multivariable calculus to differential forms?

One way to define the algebra of differential forms $\Omega(M)$ on a smooth manifold $M$ (as explained by John Baez's week287) is as the exterior algebra of the dual of the module of derivations on ...
62
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4answers
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Parallelizability of the Milnor's exotic spheres in dimension 7

Are the Milnor's seven dimensional exotic spheres parallelizable?
39
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7answers
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What are the open subsets of $\mathbb{R}^n$ that are diffeomorphic to $\mathbb{R}^n$

Hello, I would like to know if there is a known necessary and sufficient property on an open subset of $\mathbb{R}^n$ such that it is diffeomorphic to $\mathbb{R}^n$ : For example : 1) Are all ...
35
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9answers
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Classification problem for non-compact manifolds

Background It is well-known that the compact two-dimensional manifolds are completely classified (by their orientability and their Euler characteristic). I'm also under the impression that there is ...
34
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4answers
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Can cotangent bundles see exotic smooth structures?

I have two questions that are inspired by a couple of questions here on MO (referenced below), as well as by a conversation with some other grad students at a summer school. Caveat: I'm not a ...
33
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15answers
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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 ...
33
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7answers
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Is there a preferable convention for defining the wedge product?

There are different conventions for defininig the wedge product $\wedge$. In Kobayashi-Nomizu, there is $\alpha\wedge\beta:=Alt(\alpha\otimes\beta)$, in Spivak, we find $\alpha\wedge\beta:=\frac{k!l!}...
32
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3answers
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When is a submanifold of $\mathbf R^n$ given by global equations?

Let $M \subset \mathbf R^n$ be a (smooth) submanifold of dimension $d$. Under which conditions does there exist global equations defining $M$? By global equations I mean : does there exist a smooth ...
31
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9answers
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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 ...
31
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3answers
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What is the classifying space of “G-bundles with connections”

Let $G$ be a (maybe Lie) group, and $M$ a space (perhaps a manifold). Then a principal $G$-bundle over $M$ is a bundle $P \to M$ on which $G$ acts (by fiber-preserving maps), so that each fiber is a $...
31
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4answers
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Are submersions of differentiable manifolds flat morphisms?

Let $\pi \colon M\to N$ be a smooth map between real smooth manifolds. Then $C^\infty(M)$ forms a module over $C^\infty(N)$ (via pullback). Is this module flat when $\pi$ is a submersion? Recall that ...
30
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6answers
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How can you tell if a space is homotopy equivalent to a manifold?

Is there some criterion for whether a space has the homotopy type of a closed manifold (smooth or topological)? Poincare duality is an obvious necessary condition, but it's almost certainly not ...
28
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2answers
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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, ...
28
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1answer
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Strong Whitney embedding theorem for non-compact manifolds

$\newcommand{\RR}{\mathbb{R}}$The present question arises from some confusion on my part regarding the precise statement of the strong Whitney embedding theorem for non-compact manifolds. The strong ...
27
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4answers
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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, ...
27
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3answers
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Is it possible to improve the Whitney embedding theorem?

Edited to fix the example, as per Zack's suggestion. Edit 2: So it turns out that when I think 'manifold' I tend to assume the nicest possible object. As I believe is standard, I would like to ...
27
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3answers
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Embeddings of $S^2$ in $\mathbb{CP}^2$

Suppose we are given an embedding of $S^2$ in $\mathbb{CP}^2$ with self-intersection 1. Is there a diffeomorphism of $\mathbb{CP}^2$ which takes the given sphere to a complex line? Note: I suspect ...
26
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2answers
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How can we detect the existence of almost-complex structures?

Any smooth $k$-manifold $M$ comes with a well-defined map $f:M\rightarrow BGL_{k}(\mathbb{R})$ (up to homotopy) classifying its tangent bundle. Since $GL_{k}(\mathbb{R})$ deformation-retracts onto $...
24
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4answers
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Is the space of diffeomorphisms homotopy equivalent to a CW-complex?

Clarification: My question concerns the homotopy type of the space of $C^k$ diffeomorphisms with the compact-open $C^k$ topology, where $0< k \leq\infty$. I have stated my question below with $k=1$ ...
24
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2answers
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Level sets of Morse functions

Every compact two dimensional manifold admits a Morse function such that any its regular level set is at most two circles. I am interested in a generalization of that phenomenon. Does there exist a ...
24
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4answers
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What is meant by smooth orbifold?

There seems to be some confusion over what the tangent space to a singular point of an orbifold is. On the one hand there is the obvious notion that smooth structures on orbifolds lift to smooth $G$-...
24
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1answer
535 views

Diffeomorphisms of finite order not in the image of a circle action

Does there exist a closed smooth manifold $M$ and a diffeomorphism $f\colon M \to M$ such that: $f$ is isotopic to the identity, $f$ is of finite order, $f^n=ID$, and $f$ is not contained in the ...
24
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1answer
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Can you flip the end of a large exotic $\mathbb{R}^4$

Can you flip the end of a large exotic $\mathbb{R}^4$ Background Definition (Exotic $\mathbb{R}^4$): An exotic $\mathbb{R}^4$ is a smooth manifold $R$ homeomorphic but not diffeomorphic to $\mathbb{...
23
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6answers
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How to get convinced that there are a lot of 3-manifolds?

My question is rather philosophical : without using advanced tools as Perlman-Thurston's geometrisation, how can we get convinced that the class of closed oriented $3$-manifolds is large and that ...
23
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2answers
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Exotic differentiable structures on R^4?

This was going to be a comment to Differentiable structures on R^3, but I thought it would be better asked as a separate question. So, it's mentioned in the previous question that $\mathbb{R}^4$ has ...
23
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2answers
813 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 ...
23
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2answers
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Cobordism of orbifolds?

Is it possible to setup classical cobordism theory in the context of orbifolds? For example, let's consider the free abelian group generated by oriented smooth orbifolds and quotient by those which ...
22
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4answers
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Can all n-manifolds be obtained by gluing finitely many blocks?

Fix a dimension $n\geqslant 2$. Let $S= \{M_1,\ldots, M_k\}$ be a finite set of smooth compact $n$-manifold with boundary. Let us say that a smooth closed $n$-manifold is generated by $S$ if it may ...
22
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1answer
298 views

Finite limits in the category of smooth manifolds?

The category of smooth manifolds does not have all finite limits. However, it does have some limits: it has finite products, it has splittings of idempotents, and it has certain other limits if we ...
21
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2answers
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Universal property of the tangent bundle

If $X$ is a scheme (over some base scheme, but which I will ignore) its tangent bundle $T(X)$ is defined as the relative spectrum of the symmetric algebra of its sheaf of differentials. Combining the ...
21
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1answer
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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 $f^{n}(x_{0})...
21
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0answers
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Do there exist exotic 4-tori?

More precisely: are there known manifolds which are homeomorphic, but not diffeomorphic to the standard 4-torus? Are there any nice invariants distinguishing such manifolds? Related: if such a ...
20
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2answers
555 views

Is there a smooth manifold which admits only rigid metrics?

Does there exist a (finite dimensional) smooth manifold $M$, such that every Riemannian metric on $M$ has no isometries except the identity? Of course, such a manifold must not admit a diffeomorphism ...
20
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2answers
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When is a closed differential form harmonic relative to some metric?

Let $\omega$ be a closed non-exact differential $k$-form ($k \geq 1$) on a closed orientable manifold $M$. Question: Is there always a Riemannian metric $g$ on $M$ such that $\omega$ is $g$-harmonic,...
20
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4answers
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Algorithmically unsolvable problems in topology

This question is inspired by a paper by B. Poonen that appeared on the arxiv some time ago: http://arxiv.org/abs/1204.0299. The paper gives a sample of algorithmically unsolvable problems from various ...
19
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7answers
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Why should I prefer bundles to (surjective) submersions?

I hope this question isn't too open-ended for MO --- it's not my favorite type of question, but I do think there could be a good answer. I will happily CW the question if commenters want, but I also ...
19
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2answers
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Questions on J. F. Nash's answer about his errors in the proof of embedding theorem

In the interview of John Nash taken by Christian Skau and Martin Gaussen, in EMS Newsletter, September, 2015 when asked Is it true, as rumours have it, that you started to work on the embedding ...
19
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1answer
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GAGA and Chern classes

My question is as follows. Do the Chern classes as defined by Grothendieck for smooth projective varieties coincide with the Chern classes as defined with the aid of invariant polynomials and ...
19
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0answers
277 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. ...
18
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6answers
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Least number of charts to describe a given manifold

Hello, I'm wondering if there is a standard reference discussing the least number of charts in an atlas of a given manifold required to describe it. E.g. a circle requires at least two charts, and ...
18
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1answer
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Manifolds with two coordinate charts

What is an early reference for the fact that if a compact, connected $n$-manifold $M$ is covered by two open sets homeomorphic to $\mathbb{R}^n$ then $M$ is homeomorphic to $S^n$? And is it true that ...
18
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1answer
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Characteristic classes of sphere bundles over spheres in terms of clutching functions

I'm trying to understand Milnor's proof of the existence of exotic 7-spheres. Milnor finds his examples among $S^{3}$ bundles over $S^{4}$ (with structure group $SO(4)$ ). Such a bundle can be ...
18
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3answers
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Noncommutative smooth manifolds

Connes defined a noncommutative analog of a closed oriented Riemannian spin^c manifold using spectral triples. Using his definition it is unclear how to separate the smooth structure from the metric. ...
17
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7answers
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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 ...
17
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5answers
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Compactification theorem for differentiable manifolds ?

Just parallelling this question, that seemed not to admit an easy answer at all, let's "soft down" the category and ask the same thing in the case of $\mathcal{C}^{\infty}$-differentiable manifolds [...
17
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4answers
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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 ...
17
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1answer
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What's the Kirby Diagram of a universal $\mathbb{R}^4$?

What's the Kirby diagram of a universal $\mathbb{R}^4$? Background Define $\mathcal{R}$ as the set of smoothings of $\mathbb{R}^4$. For two oriented elements $R_1$, $R_2$ in $\mathcal{R}$ we can ...
17
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1answer
548 views

Differential Topology over $\mathbb{Q}$

I have a specific question in mind, but it requires some explanation and context before it can be formally stated. To summarize it in a sentence, this is it: Are every two rational manifolds of the ...
17
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1answer
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How are these algebraic and geometric notions of homotopy of maps between manifolds related?

Let $M$ and $N$ be smooth manifolds, and $f,g: M \to N$ smooth maps. Denote by $(\Omega^\bullet M,\mathrm d_M)$ and $(\Omega^\bullet N, \mathrm d_N)$ the cdgas of de Rham forms in each manifold, and ...
16
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1answer
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Is there a notion of a chain complex with corners?

Roughly speaking, algebraic topology works by reducing questions about topological objects such as manifolds and cell to questions about chain complexes. On the topological side, although in the PL ...