Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.

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75
votes
15answers
9k views

What is torsion in differential geometry intuitively?

Hi, given a connection on the tangent space of a manifold, one can define its torsion: $$T(X,Y):=\triangledown_X Y - \triangledown_Y X - [X,Y]$$ What is the geometric picture behind this ...
17
votes
8answers
2k views

Determining a surface in $\mathbb{R}^3$ by its Gaussian curvature

A curve in the plane is determined, up to orientation-preserving Euclidean motions, by its curvature function, $\kappa(s)$. Here is one of my favorite examples, from Alfred Gray's book, Modern ...
5
votes
2answers
422 views

Cotangent bundle lift theorem

Let $M$ be a smooth manifold and $T^\ast M$ be its cotangent bundle. Consider the tautological 1-form $\theta$ on $T^\ast M$ ($\theta=\sum y_i dx^i$ in local canonical coordinate systems). A ...
2
votes
1answer
491 views

Is it possible to define a structure of differentiable manifold on the Hilbert cube $[0,1]^\mathbb{N}$?

Is it possible to define a structure of differentiable (smooth) manifold on the Hilbert cube $[0,1]^\mathbb{N}$ ? Has it been done in the literature? In textbooks, only the Banach case is treated, ...
92
votes
16answers
12k views

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 ...
38
votes
7answers
4k views

Riemannian surfaces with an explicit distance function?

I'm looking for explicit examples of Riemannian surfaces (two-dimensional Riemannian manifolds $(M,g)$) for which the distance function d(x,y) can be given explicitly in terms of local coordinates of ...
20
votes
3answers
2k views

Can every manifold be given an analytic structure?

Let $M$ be a (real) manifold. Recall that an analytic structure on $M$ is an atlas such that all transition maps are real-analytic (and maximal with respect to this property). (There's also a sheafy ...
11
votes
5answers
2k views

Why do I need densities in order to integrate on a non-orientable manifold?

Integration on an orientable differentiable n-manifold is defined using a partition of unity and a global nowhere vanishing n-form called volume form. If the manifold is not orientable, no such form ...
9
votes
4answers
2k views

How to see the Phase Space of a Physical System as the Cotangent Bundle

Two things today motivated this question. First, the professor said that in a lecture Thurston mentioned Any manifold can be seen as the configuration space of some physical system. Clearly we ...
15
votes
2answers
830 views

Converse to Stokes' Theorem

Does satisfying Stokes' Theorem imply that a form is linear? Let $M$ be an $n$-manifold. A differential $k$-form $\omega \in \Omega^k M$ assigns to each point $x \in M$ a function $\omega_x : ...
23
votes
3answers
2k views

Trapped rays bouncing between two convex bodies

At some point during my research I was confronted with this problem, but I did not dedicate serious time to it. Anyway it stayed in the back of my mind and I'm still interested in hints for it. ...
12
votes
3answers
837 views

How transitive are the actions of symplectomorphism groups ?

This question is motivated by the classical fact from differential geometry : Let $M$ be a smooth manifold of dimension at least $2$. Then for any $n$ the diffeomorphism group $\textrm{Diff}(M)$ ...
8
votes
2answers
331 views

Surface in 3D that realizes all pairs of principal curvatures

This is a question that Willie Wong raised in comments after he answered my question, Surface analog of clothoid: curvatures covering $\mathbb{R}^3$. Willie's question is more interesting (and ...
4
votes
0answers
222 views

Symmetric matrices and Hilbert's fourth problem

From the analytic viewpoint, the Busemann-Pogorelov solution of Hilbert's fourth problem is summarized in the following result: Theorem. All straight lines are extremals of the variational problem $$ ...
3
votes
1answer
218 views

Explicit form for hermitian structure $h$ with respect to $\omega$

Let $(M,\omega)$ be a symplectic manifold. and $\pi:L\to M$ be a complex line bundle , we denote $h$ as hermitian structure,i.e. if $s,s'$ are smooth sections of $L$ and if $X$ is a vector field on ...
3
votes
1answer
254 views

Existence and Properties of 3D Curves with unusual 2D $(\kappa(s),\tau(s))$ Trajectories

This question is inspired by Surface in 3D that realizes all pairs of principal curvatures While one can imagine, that a 3D surface could exist, that realizes all pairs of principal curvatures, ...
1
vote
1answer
249 views

Projectively equivalent connections

We can define projective structure on a manifold in two ways. First we can define it as a maximal atlas of charts from the open subsets of manifold to the projective space , such that transitions maps ...
1
vote
1answer
294 views

Decomposition of hermitian form used in the definition of Griffiths/Nakano positivity

Let $E$ be a hermitian holomorphic vector bundle over a complex manifold $X$. Then $\Theta(E)$, the curvature of $E$, is a section of $\bigwedge^{1,1}X\otimes\operatorname{End}(E)$. However, we have ...
58
votes
5answers
3k views

Is there a sheaf theoretical characterization of a differentiable manifold?

I'm going through the crisis of being unhappy with the textbook definition of a differentiable manifold. I'm wondering whether there is a sheaf-theoretic approach which will make me happier. In a ...
51
votes
15answers
7k views

Are there examples of non-orientable manifolds in nature?

Whilst browsing through Marcel Berger's book "A Panoramic View of Riemannian Geometry" and thinking about the Klein bottle, I came across the sentence: "The unorientable surfaces are never discussed ...
51
votes
6answers
4k views

Is there an analogue of curvature in algebraic geometry?

I am not an expert, but there seems to be an enormous technical difference between algebraic geometry and differential/metric geometry stemming from the fact that there is apparently no such thing as ...
46
votes
10answers
11k views

why study Lie algebras?

I don't mean to be rude asking this question, I know that the theory of Lie groups and Lie algebras is a very deep one, very aesthetic and that has broad applications in various areas of mathematics ...
56
votes
6answers
5k views

Is there an underlying explanation for the magical powers of the Schwarzian derivative?

Given a function $f(z)$ on the complex plane, define the Schwarzian derivative $S(f)$ to be the function $S(f) = \frac{f'''}{f'} - \frac{3}{2} (\frac{f''}{f'})^2$ Here is a somewhat more conceptual ...
49
votes
13answers
4k views

What makes four dimensions special?

Do you know properties which distinguish four-dimensional spaces among the others? What makes four-dimensional topological manifolds special? What makes four-dimensional differentiable manifolds ...
38
votes
2answers
3k views

Cohomology and fundamental classes

Let X be a real orientable compact differentiable manifold. Is the (co)homology of X generated by the fundamental classes of oriented subvarieties? And if not, what is known about the subgroup ...
61
votes
11answers
8k views

Is it possible to capture a sphere in a knot?

You and I decide to play a game: To start off with, I provide you with a frictionless, perfectly spherical sphere, along with a frictionless, unstretchable, infinitely thin magical rope. This rope ...
41
votes
4answers
2k views

Random manifolds

In the world of real algebraic geometry there are natural probabilistic questions one can ask: you can make sense of a random hypersurface of degree d in some projective space and ask about its ...
23
votes
5answers
3k views

A geometric interpretation of the Levi-Civita connection?

Let $M$ be a Riemannian manifold. There exists a unique torsion-free connection in the (co)tangent bundle of $M$ such that the metric of $M$ is covariantly constant. This connection is called the ...
14
votes
4answers
1k views

What is a section?

This question comes out of the answers to Ho Chung Siu's question about vector bundles. Based on my reading, it seems that the definition of the term "section" went through several phases of ...
9
votes
6answers
2k views

CW-structures and Morse functions: a reference request

The following is probably well known, but I wasn't able to locate a reference in the literature. Let $f$ be a Morse function on a smooth compact manifold $M$ without boundary and let $\rho$ be a ...
28
votes
3answers
2k views

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 ...
22
votes
5answers
3k views

References for “modern” proof of Newlander-Nirenberg Theorem

Hi, I'm starting to prepare a graduate topics course on Complex and Kahler manifolds for January 2011. I want to use this course as an excuse to teach the students some geometric analysis. In ...
18
votes
4answers
3k views

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 ...
17
votes
2answers
1k views

Does it make sense to talk about smooth bundles of Hilbert spaces?

Is there a notion of "smooth bundle of Hilbert spaces" (the base is a smooth finite dimensional manifold, and the fibers are Hilbert spaces) such that: 1• A smooth bundle of Hilbert spaces ...
15
votes
5answers
2k views

Weitzenböck Identities

I asked this question at Maths Stack Exchange, but I haven't received any replies yet (I'm not sure how long I should wait before it is acceptable to ask here, assuming there is such a period of ...
23
votes
2answers
808 views

Is there a convenient differential calculus for cojets?

I understand the basics of exterior differential geometry and how to do calculus with exterior differential forms. I know how to use this to justify the notation dy/dx as a literal ratio of the ...
22
votes
5answers
2k views

Two definitions of Calabi-Yau manifolds

Why is it that the vanishing of the integral first Chern class of a compact Kahler manifold is equivalent to the canonical bundle being trivial? I can see that it implies that the canonical bundle ...
28
votes
5answers
2k views

Is there a “geometric” intuition underlying the notion of normal varieties?

I first got concious of the notion of normal varieties around 3 years ago and despite the fact that by now I can manipulate with it a bit, this notion still puzzles me a lot. One thing that strikes me ...
19
votes
5answers
801 views

Lightray trapped between two mirror disks: Computation formulation?

I would like to calculate the angle of a ray $r$ from a given point $p$ such that it gets "stuck" reflecting between two congruent mirror-disks. For why there is such a ray, see the (amazing!) answer ...
19
votes
1answer
754 views

Are the only local minima of $\angle(v, Av)$ the eigenvectors?

Let $A$ be an invertible $n \times n$ complex matrix. For $v \in \mathbb{CP}^{n-1}$, define $$d(v) = \frac{|\langle A \tilde{v}, \tilde{v} \rangle |^2}{ \langle A \tilde{v}, A \tilde{v} \rangle ...
9
votes
2answers
988 views

When does the tangent bundle of a manifold admit a flat connection?

Let $M$ be a smooth manifold, and let $TM$ denote its tangent bundle. Under what conditions does $TM$ admit a flat connection $\omega$? Edit: Formerly, I asked about a flat connection on the frame ...
22
votes
3answers
794 views

Rep Theory Consequences of Bott--Weil--Borel

I've been getting interested in the (Bott--)Borel--Weil theorem lately. As a (mainly) geometer it is very interesting to see representation appearing (from nowhere as far as I can see) in the theory ...
12
votes
3answers
656 views

What is the DGLA controlling the deformation theory of a complex submanifold?

Let $X$ be a complex manifold, $Y\hookrightarrow X$ a complex compact submanifold. Let $T_{X/Y}$ denote the normal bundle of $Y$ in $X$, and $\mathcal{O}(T_{X/Y})$ its sheaf of holomorphic sections. ...
10
votes
3answers
497 views

Characterizing Hessians among symmetric bilinear tensors

I apologize in advance if this is somewhat elementary, but: Let $(M,g)$ be a compact Riemannian manifold. Is there a "characterization" of which symmetric bilinear tensors $B\in Sym^2(M)$ are ...
9
votes
1answer
2k views

Derivative of Exponential Map

Given a Riemannian manifold $M$, let $\gamma: (a,b) \to M$ be a geodesic and $E$ a parallel vector field along $\gamma$. Define $\varphi: (a,b) \to M$ by $t \mapsto \exp_{\gamma(t)}(E(t))$. Is there a ...
18
votes
1answer
757 views

Under exactly what (extra) conditions (if any) is a connected Hausdorff manifold with a Riemannian metric a metric space?

The setting is that manifolds are Banach manifolds, not necessarily finite dimensional. No other assumption is made about the topology of the manifold. In particular, it is not assumed to be regular ...
18
votes
0answers
1k views

Ricci flat metric on $n$-sphere?

Can you put a Ricci flat metric on the $n$-sphere, $n>4$?
15
votes
3answers
2k views

How to Tackle the Smooth Poincare Conjecture

The last remaining problem in this whole "everything is a sphere" business, is the Smooth Poincare Conjecture in dimension 4: If $X\simeq_\text{homo.eq.} S^4$ then $X\approx_\text{diffeo} S^4$. ...
13
votes
2answers
1k views

Switching from pure mathematics (e.g. geometry) to more applied areas (e.g imaging) after Ph.D., as postdoc and chance of getting such a postdoc?

Before I start my question, I should probably mention that this question might not be the right question to ask here, but I tried academiabeta, and stackoverflow, but without getting any to-the-point ...
13
votes
2answers
722 views

Is there a unified reason that there are an infinite number of geodesics between nonconjugate points on a compact manifold?

The proof of this statement seems to break into two really different arguments. So, I'm wondering if there is a better argument that can explain them both, or whether it's really just two theorems ...