The tag has no wiki summary.

learn more… | top users | synonyms (1)

17
votes
2answers
1k views

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 ...
13
votes
2answers
752 views

Nice application of generalized smooth spaces

I am a fan of category theory in general, and I appreciate that various brands of generalized smooth spaces (Diffeological spaces, Chen spaces, Frolicher spaces ...) form much nicer categories of ...
6
votes
3answers
498 views

Can homologous submanifolds be connected by an immersed manifold with boundary?

Supposed I have an n-dimensional manifold M with a k-dimensional submanifold that is homologous to zero (or, equivalently, two homologous submanifolds). Can I always construct a k+1-dimensional ...
2
votes
2answers
2k views

On a proof of the existence of tubular neighborhoods.

Studying analysis on manifolds, I have found, in the proof of the existence of tubular neighborhoods, a reference to theorem 3.1.2 in "Topologie algebrique et theorie des faisceaux" of Godement. ...
5
votes
2answers
771 views

Is volume--preserving an intrinsic property?

Let $M$ be a compact smooth manifold without boundary. A Riemannian metric $g$ on $M$ induces a volume measure (or Lebesgue measure) $m_g$ on $M$. A diffeomorphism $f:M\to M$ is said to be ...
19
votes
6answers
3k views

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 ...
16
votes
1answer
790 views

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 ...
0
votes
0answers
467 views

A doubt on a problem in Manifolds, Tensor Analysis and Applications

Having tried to solve exercise 4.4-7 to have another proof of Frobenius Theorem, I would ask you a question. This is what I have understood: In Step 2 there is to prove, for any tangent subbundle ...
1
vote
1answer
225 views

Commutativity of pullbacks and the exterior derivative as an unbounded operator on $L^2$

Let $d_c, \delta_c$ be operators with domains $D(d_c) = D(\delta_c) = C_{c}^\infty(\wedge T^\ast M)$. We let $d_c$ be the usual exterior derivative on compactly supported smooth forms, ie., $d_c\omega ...
1
vote
1answer
266 views

why rotation group is a smooth 3-manifold [closed]

Do you need the implicit function theorem? Can you give an explicit charts?
12
votes
1answer
491 views

smoothly varying smooth structures

Can one vary smooth structures on $\mathbb R^4$ smoothly/continuously? This question popped out of Ben's answer here.
7
votes
2answers
630 views

Fibrewise homotopy-equivalence of unit sphere bundles vs isomorphism of tangent bundles

Let $M$ be a smooth $m$-dimensional manifold, $TM$ its tangent bundle and $SM$ its unit sphere bundle. Are there some simple examples where $SM$ is fibrewise homotopy-equivalent to the trivial ...
3
votes
0answers
215 views

Density of C^\infty in the domain of the exterior derivative on a noncompact, complete manifold?

Let $(M,g)$ be a geodesically complete Riemannian manifold that is not necessarily compact. Futhermore, assume that $M$ has at most exponential volume growth (ie., locally doubling property). Let ...
1
vote
3answers
620 views

analytic structure on lie groups

I need a reference for a result I have heard only very vaguely "A lie group (smooth) has a compatible analytic manifold structure". (Would even appreciate a concise way to refer to the result..) I ...
11
votes
2answers
944 views

Smooth structures on the connected sum of a manifold with an Exotic sphere

What can we say about the connected sum of a manifold $M^n$ with an Exotic sphere? Is is possible some of them are still diffemorphic to $M^n$. Is it possible to classifying all the exotic smooth ...
3
votes
2answers
1k views

A metric for Grassmannians

Hello everybody! I'm reading an article by Ricardo Mane, Hausdorff dimension is dipheomorphism. I'm having a technical problem. Sorry for my ignorance but Would you like a references where I can find ...
12
votes
3answers
690 views

orientations for zero-dimensional manifolds

I am teaching a course on manifolds, and soon I will have to prove the Stokes' theorem which, of course, involves defining oriented manifolds. There are many ways to define an oriented manifold. My ...
5
votes
2answers
488 views

Definition of the Kervaire invariant for normal maps (as in Browder's book)

Browder's book "Surgery on simply-connected manifolds" defines the Kervaire invariant in a very general setting. My question is: how does one get the more usual definition of the invariant for a ...
2
votes
2answers
964 views

When are diffeomorphisms from a manifold to itself homotopic to the identity?

I'm sure people in the field know this, but I'm not in the field. Under what conditions (be they on the manifold or the map) is a diffeomorphism from a differentiable manifold $M$ to itself homotopic ...
9
votes
5answers
1k views

Ricci Curvature in Infinite Dimensions?

Is there a good notion of "Ricci curvature" in infinite dimensions? My intuitive understanding of Ricci curvature is that it is some kind of an "average" of the curvature tensor over "different ...
1
vote
1answer
381 views

Understanding manifold GL+(3,R)/SO(3) ?

I'm trying to better understand the manifold GL+(3,R)/S0(3) which is diffeomorphic to positive definite symmetric matrices. My motivation is to understand U in F = RU where F in GL+(3,R) = deformation ...
21
votes
4answers
1k views

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 ...
7
votes
2answers
1k views

Slice knots and exotic $\mathbb R^4$

In the http://arxiv.org/abs/math/0606464v1 I read "If you want to prove existence of exotic smooth structure on $\mathbb R^4$ you can do this if you are in possession of a knot which is ...
12
votes
2answers
426 views

Self homeomorphisms of $S^2\times S^2$

Every matrix $A\in SL_2(\mathbb{Z})$ induces a self homeomorphism of $S^1\times S^1=\mathbb{R}^2/\mathbb{Z}^2$. For different matrices these homeomorphisms are not homotopic, as the induced map on ...
5
votes
0answers
135 views

What is the name of the quotient of the Stiefel manifold of $k$-frames by the symmetric group of $k$ letters?

Let $V_k(\mathbb{R}^n)$ be the Stiefel manifold of $k$-frames in $\mathbb{R}^n$. The symmetric group of $k$ letters $\Sigma_k$ acts freely by permuting vectors in $k$-frames. Does the quotient ...
4
votes
2answers
643 views

In Diff, are the surjective submersions precisely the local-section-admitting maps?

Question as in title (Diff = category of smooth manifolds and smooth maps) I thought I'd convinced myself this is true, so this is just a sanity check. Also, what about for settings other than ...
4
votes
1answer
609 views

Studying non-linear PDEs with manifolds

I'm sorry if this is an inappropriate forum to ask this question on, for I fear it is pretty undergraduate-level one :) I was contemplating on the study of non-linear PDEs. Is it possible to reduce a ...
9
votes
2answers
571 views

Colimits of manifolds

This question tells us that in general colimits do not exist in the category of manifolds. However, this negative answer is not very satisfying. A manifold can be considered as a colimit of its ...
5
votes
1answer
860 views

Gluing two diffeomorphisms together

A fundamental construction in a first course on manifolds is to build a smooth function $\psi\colon \mathbb{R} \to \mathbb{R}$ with the property that for some $0<\delta<\epsilon$ we have ...
5
votes
1answer
667 views

Exotic spheres detected in higher homotopy

Thinking about exotic 7-spheres, one can look at the maps $\cdots \rightarrow \Omega^2Diff(D^4, rel \space \partial) \rightarrow \Omega Diff(D^5, rel \space \partial) \rightarrow Diff(D^6, rel \space ...
6
votes
2answers
959 views

Totally geodesic surfaces in fibered 3-manifolds

Is there an easy example of a (closed) hyperbolic 3-manifold that fibers over the circle but contains some totally geodesic surface? (Of course such manifolds exist if the 'Virtually Fibered ...
3
votes
2answers
304 views

Extending diffeomorphisms of Riemannian surfaces to the ambient space

Question 1: Given a smooth Riemannian surface $M$ in $R^3$ (i.e., a smooth Riemannian 2-manifold embedded in $R^3$) and a diffeomorphism $f: M\rightarrow M$ of class $C^{k\geq 2}$, does $f$ admit a ...
21
votes
14answers
6k 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: 1) the result that the solution space of a non-degenerate system of equations ...
0
votes
1answer
131 views

quasi-separated manifolds

Let $M$ be a smooth manifold. Let's call $M$ quasi-seperated if $M$ has the following property: If $B,C \subseteq M$ are open balls, then $B \cap C \subseteq M$ is a finite(!) union of open balls. By ...
9
votes
2answers
659 views

On the $\mathbb R$-algebra structure on $C^\infty(M)$.

As it is more or less well-know, and as it has come up on MO a couple of times, the $\mathbb R$-algebra $C^\infty(M)$ of smooth functions on a (say) compact manifold contains essentially everything ...
9
votes
2answers
532 views

The ring $C^{\infty}(M)$?

Let $M$ be a smooth paracompact manifold. I think that the ring $C^{\infty}(M)$ contains many (possibly almost all?) geometric or topological information about $M$. (e.g. Let $E$ be a vector bundle ...
7
votes
1answer
850 views

How well can we localize the “exoticness” in exotic R^4?

My question concerns whether there is a contradiction between two particular papers on exotic smoothness, arxiv:0807.4248v1 and arxiv:gr-qc/9404003v1. The former asserts: "Let $M$ be a smooth closed ...
3
votes
2answers
994 views

Is it true that exotic smooth R^4 cannot be diffeomorphic to RxN, where N is a 3-manifold?

Since $\mathbb{R}$ and any 3-manifold $N$ must be non-exotic, their product $\mathbb{R}\times N$ cannot possibly be diffeomorphic to exotic $\mathbb{R}^4$, correct? Update: Andy Putman already ...
7
votes
1answer
464 views

Is every graded manifold affine, and is this definition of graded manifold the right one?

The following definition is from: Dmitry Roytenberg, "AKSZ-BV formalism and Courant algebroid-induced topological field theories", Letters in Mathematical Physics, 2007 vol. 79 (2) pp. 143-159, ...
2
votes
2answers
547 views

exotic smooth structure clarification

Does the existence of exotic smooth structure in $\mathbb{R}^4$ imply the existence of an atlas which has a $C^0$ mapping to the Cartesian atlas, but not a $C^k$ mapping (for some finite $k$)? Does ...
4
votes
1answer
2k views

Hessian as a tensor, multi-dimensional taylor series, and generalizations

The Hessian matrix $\{\partial_i \partial_j f \}$ of a function $f:\mathbb{R}^n \to \mathbb{R}$ depends on the coordinate system you choose. If $x_1,\cdots,x_n$ and $y_1,\cdots,y_n$ are two sets of ...
21
votes
2answers
2k views

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 ...
10
votes
5answers
784 views

Which manifolds admit a diffeomorphism of order $n$?

Let $n>1$. Which smooth manifolds admit a diffeomorphism $f$ of order $n$? For a closed orientable surface $S_g$ of genus $g$ and $n=2$ the answer is in the affirmative, since $S_g$ can be ...
5
votes
1answer
412 views

When does a submersion have connected fibers?

Can we characterize when a submersion $F:M \to N$ between two smooth manifolds has connected fibers? If this is too hard, what are some sufficient conditions?
6
votes
3answers
753 views

Checking whether the image of a smooth map is a manifold

I have a specific problem, but would also like to know how to tackle the general case. I will first state the genral question. Let $M$ be an embedded submanifold of $\mathbb{R}^n$ and let $F: ...
4
votes
1answer
342 views

Do Lie algebroids pull back (along submersions)?

There are more general definitions, but for my purposes a Lie algebroid on a smooth manifold $X$ is a vector bundle $A \to X$, a map $\rho: A \to {\rm T}X$ of vector bundles over $X$, and a bracket ...
10
votes
3answers
887 views

smooth sections of smooth fiber bundles

A maybe trivial question about fiber bundles (I'm not an expert, and I didn't find quickly an answer looking here and there). Suppose you are given fiber bundle $p\colon E\to M$, where $E,M$ are ...
13
votes
2answers
2k views

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 ...
3
votes
2answers
239 views

A local transitivity property of the automorphism group of a foliated manifold

Let $(M,\mathcal F)$ be a smooth foliated manifold. An automorphism of $(M,\mathcal F)$ is a diffeomorphism of $M$ that takes leaves of $\mathcal F$ onto leaves. Let now $L$ be a leaf of $\mathcal F$. ...
16
votes
5answers
1k views

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 ...