Questions tagged [dg.differential-geometry]
Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
8,907 questions
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Why is there no symplectic version of spectral geometry?
First, recall that on a Riemannian manifold $(M,g)$ the Laplace-Beltrami operator $\Delta_g:C^\infty(M)\to C^\infty(M)$ is defined as
$$
\Delta_g=\mathrm{div}_g\circ\mathrm{grad}_g,
$$
where the ...
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Stokes theorem for manifolds with corners?
Maybe this is an elementary question, but I'm unable to find the appropriate reference for it. The Stokes theorem tells us that, for a $n+1$-dimensional manifold $M$ with boundary $\partial M$ and any ...
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Example of 4-manifold with $\pi_1=\mathbb Q$
This might be well known for algebraic topologist. So I am looking for an explicit example of a 4 dimensional manifold with fundamental group isomorphic to the rationals $\mathbb Q$.
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Is there any way to rewrite a partial differential equation using language of differential forms, tensors, etc?
My question is: usually, a partial differential equation, for example, those coming from physics, is written in a language of vector calculus in a local coordinate. Is there any way (or any algorithm) ...
- 1,499
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Rolle's theorem in n dimensions
This looks like a statement from a calculus textbook, which perhaps it should be.
"Rolle's theorem". Let $F\colon [a,b]\to\mathbb R^n$ be a continuous function such that $F(a)=F(b)$ and $F'(t)$ ...
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2
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Probing a manifold with geodesics
Supposed you stand at a point $p \in M$ on a smooth 2-manifold $M$
embedded in $\mathbb{R}^3$.
You do not know anything about $M$.
You shoot off a geodesic $\gamma$ in some direction $u$,
and learn ...
- 151k
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What is special to dimension 8?
Dimension $8$ seems special, as the partial list below might indicate.
Is there any overarching reason that dim-$8$ is "more special" than, say, dim-$9$?
Surely it isn't it, in the end, simply because ...
28
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What is the relationship between various things called holonomic?
The following things are all called holonomic or holonomy:
A holonomic constraint on a physical system is one where the constraint gives a relationship between coordinates that doesn't involve the ...
- 16k
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What algebraic structure characterizes all natural operations between differential operators and differential forms?
On a smooth manifold $M$ one can define various algebraic structures, natural with respect to diffeomorphisms:
the differential graded-commutative algebra $\Omega(M)$ of differential forms on $M$;
...
- 37.8k
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Examples of Kan extensions, adjunctions, and (co)monads in analysis, Lie theory, and differential geometry?
In introductory texts on category theory, it seems like the majority of examples come from algebraic topology, algebra, and logic.
What are some good examples of Kan extensions, adjunctions, and (co)...
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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 ...
27
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5
answers
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Algebraic description of compact smooth manifolds?
Given a compact smooth manifold $M$, it's relatively well known that $C^\infty(M)$ determines $M$ up to diffeomorphism. That is, if $M$ and $N$ are two smooth manifolds and there is an $\mathbb{R}$-...
- 6,546
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Does a Riemannian manifold with bounded geometry admit an isometric proper embedding into Euclidean space with uniformly thick tubular neighborhood?
Suppose $(M,g)$ is an open Riemannian manifold with bounded geometry, i.e., the injectivity radius is $\ge \epsilon>0$ and each iterated covariant derivative of curvature is bounded with respect to ...
- 25.3k
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The Chern-Simons/Wess-Zumino-Witten correspondence
I have often seen a relationship being alluded to between these two theories but I am unable to find any literature which proves/derives/explains this relationship.
I guess in the condensed matter ...
- 3,541
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Intuition for symplectic groups
My question essentially breaks down to
How do you, a working mathematician, think about (real) symplectic groups? How do you visualize symplectic (linear) transformations? What intuition do you ...
- 779
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Is there a Chern-Gauss-Bonnet theorem for orbifolds?
There's a Gauss-Bonnet theorem for compact 2-orbifolds(due to Satake, I think), which gives a relation between the curvature of a Riemannian orbifold and the orbifold topology(i.e. taking into account ...
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26
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Intuition for mean curvature
I would like to get some intuitive feeling for the mean curvature. The mean curvature of a hypersurface in a Riemannian manifold by definition is the trace of the second fundamental form.
Is there ...
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Undergraduate differential geometry texts
Can anyone suggest any basic undergraduate differential geometry texts on the same level as Manfredo do Carmo's Differential Geometry of Curves and Surfaces other than that particular one?
(I know a ...
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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 ...
- 541
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Which Kahler Manifolds are also Einstein Manifolds?
Is it known which Kahler manifolds are also Einstein manifolds? For example complex projective spaces are Einstein. Are the Grassmannians Einsein? Are all flag manifolds Einstein?
- 1,479
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Does Ricci flow depend continuously on the initial metric?
Consider a version of Ricci flow for which short time existence and uniqueness are known,
e.g. the Ricci flow on a closed manifold. Does the solution $g_t$ for small $t$ depend continuously on the ...
- 29.1k
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how do you visualize characteristic class?
For cohomology, there are some equivalent definitions when the object we consider is sufficiently nice. Since I mainly work with algebraic variety, I will restrict the objects I am considering to be ...
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Book Recommendation - PDE's for geometricians / topologists
I am looking for recommendations for a book on partial differential equations, which is not written for applied mathematicians but rather focused on geometry and applications in topology, as well as ...
26
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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 ...
- 151k
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Loop Spaces as Generalized Smooth spaces or as Infinite dimensional Manifolds?
There are two ways to define smooth mapping spaces and I want to know how they compare.
Let's take the concrete special case of free loops spaces. I think this is the most studied example so will ...
- 27.5k
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Does Ricci flow with surgery come from sections of a smooth Riemannian manifold?
More precisely, is Ricci flow with surgery on a 3-dimensional Riemannian manifold M given by the "constant-time" sections of some canonical smooth 4-dimensional Riemannian manifold?
There would be a ...
- 20.7k
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Geometric interpretation of Cartan's structure equations
Given a linear connection on a Riemmanian manifold $M$ and $\phi^1,...,\phi^n$ a local frame for $T^*M$ we can define the connection 1-forms $\omega^j_i$. We define the curvature 2-forms by $\Omega_i^...
- 2,035
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Why is the half-torus rigid?
The half-torus surface that results from slicing a torus like a bagel,
depicted below (left), is isometrically rigid.
I know this from a remark of Alexandrov in
Mathematics: Its ...
- 151k
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Harmonic spinors on closed hyperbolic manifolds
Does anyone know an example of a closed spin hyperbolic manifold of dimension 3 or greater such that the kernel of the Dirac operator is non-trivial?
I'm mainly interested in the 3-dimensional case ...
- 19.4k
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Euler characteristic and universal cover
Let $M$ be a compact manifold, let $\tilde{M}$ be its universal cover, and suppose that the Euler characteristic $\chi(\tilde{M})=0$.
My question is: does this imply that $\chi(M)=0$?
This is clear if ...
- 1,452
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Vector fields on $(4n+1)$-spheres
If $n$ is odd then $S^{n-1}$ doesn't admit a nowhere-vanishing vector field, and if $n$ is even then there does exist one (Hairy Ball Theorem). We can then ask, on $S^{n-1}$, what is the maximum ...
- 17.5k
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Examples of Einstein four-manifolds of negative sectional curvature
Are there any nontrivial compact Einstein four-manifolds of negative or nonpositive sectional curvature? by nontrivial we mean not quotients of $\mathbb{H}^4$, $\mathbb{C}H^2$, $\mathbb{H}^2\times\...
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What is a square root of a line bundle?
If ${L}$ is a line bundle over a complex manifold, what does the square root line bundle $L^{\frac{1}{2}}$ mean?
After some google, I got to know that there are certain conditions for the existence of ...
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Integration and Stokes' theorem for vector bundle-valued differential forms?
Is there a version of Stokes' theorem for vector bundle-valued (or just vector-valued) differential forms?
Concretely: Let $E \rightarrow M$ be a smooth vector bundle over an $n$-manifold $M$ equipped ...
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Is it possible for a metric on a smooth manifold to be smooth?
Are there any smooth manifolds $M$ with the following property:
There exist a realizing metric $d$ (i.e $d$ induces the topology on $M$), and $d$ is smooth on all of $M \times M$?
If not, is it ...
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How to understand Chern-Simons action
Hi all. The question I have should be a rather simple one, but I just can't think it through.
So the Chern-Simons action reads
\begin{equation}
S = \int_M {\rm tr} (A\wedge dA + \frac{2}{3} A\wedge A ...
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If the total Chern class of a vector bundle factors, does it have a sub-bundle?
Motivation: $T_{\mathbb P^2}$ isn't an extension of line bundles
Here's a trick to show that the tangent bundle $T$ of $\mathbb P^2$ is not an extension of line bundles. If it were, we would have a ...
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Isometric embedding of SO(3) into an euclidean space
Consider $SO(3)$ with its bi-invariant metric and $R^n$ the euclidean space of dimension $n$. What is the minimal value of $n$ such that there exists an isometric embedding $f: SO(3) \to R^n$?
- 251
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Functional approach vs jet approach to Lagrangian field theory
Context: I am a PhD student in theoretical physics with higher-than-average education on differential geometry. I am trying to understand Lagrangian and Hamiltonian field theories and related concepts ...
- 2,792
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Does every large $\mathbb{R}^4$ embed in $\mathbb{R}^5$?
This question was prompted by my answer to this question.
An exotic $\mathbb{R}^4$ is a smooth manifold homeomorphic to $\mathbb{R}^4$ which is not diffeomorphic to $\mathbb{R}^4$ with its standard ...
- 19.3k
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Ellipse naturally associated with a polygon
My colleagues and I have stumbled onto a way to associate an ellipse, or equivalently a positive definite symmetric matrix, to a polygon that is different from other better known ways. We want to know ...
- 27.5k
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Are there Ricci-flat riemannian manifolds with generic holonomy?
This may well be an open problem, I'm not sure.
In Berger's classification (refined by Simons, Alekseevsky, Bryant,...) of the holonomy representations of irreducible non-symmetric complete simply-...
- 33.3k
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Do free higher homotopy classes of compact Riemannian manifolds have preferred representatives?
A well known theorem of Cartan states that every free homotopy class of closed paths in a compact Riemannian manifold is represented by a closed geodesic (theorem 2.2 of Do Carmo, chapter 12, for ...
- 29.2k
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Smooth manifolds as idempotent splitting completion
The nlab has a particularly interesting thing to say about the category of smooth manifolds: it is the idempotent-splitting completion of the category of open sets of Euclidean spaces and smooth maps.
...
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Relationship between Green's function and geodesic distance?
I am interested in showing that a certain Green's function can be used to approximate the distance function on a Riemannian manifold in the following sense. Let $(M,g)$ be a Riemannian manifold and ...
- 3,059
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Alternate proofs that hyperbolic plane can’t be isometrically immersed in $\mathbb{R}^3$
A famous theorem of Hilbert says that there is no smooth immersion of the hyperbolic plane in 3-dimensional Euclidean space. The expositions of this that I know of (in eg do Carmo’s book on curves/...
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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 ...
- 2,754
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Interplay between Loop Quantum Gravity and Mathematics
It is known that there are many interesting connections between String Theory and modern Mathematics, with a rich feedback going on in both directions: there have been advances in mathematics thanks ...
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Can the constant rank theorem for smooth manifolds be generalized to nonconstant rank?
The constant rank theorem says that
if $f\colon M→N$ is a smooth map whose rank equals some fixed $k≥0$ at any point of $M$, then, locally with respect to $M$ and $N$, the map $f$ assumes the easiest ...
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