Questions tagged [dg.differential-geometry]

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

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Mountain Pass Theorem on a non-vector space

(Reposted from Math StackExchange because this may be more suited for professional mathematicians.) The Mountain Pass Theorem roughly says the following. Consider a differentiable functional $I: H \...
Sebastian's user avatar
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7 votes
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330 views

A characterization of flat metrics via global vector fields

Let $(M,g)$ be a Riemannian manifold with $LC$ conncection $\nabla$. Assume that for every three global vector fields $X,Y,Z \in \chi^{\infty}(M)$ with $[X,Y]=0$ we have $\nabla_{X} \nabla_{...
Ali Taghavi's user avatar
3 votes
1 answer
507 views

De Turck trick on mean curvature flow

I am reading the book Lecture on mean curvature flow by Xi-Ping Zhu. Suppose $M^n$ is an n-dimension smooth manifold and $X(x,t):M^n \rightarrow R^{n+1}$ be a one-parameter family of smooth immersion....
mnmn1993's user avatar
6 votes
1 answer
481 views

Yau-Uhlenbeck inequality works for higher Chern class?

If the holomorphic vector bundle of $E$ on a Kähler manifold $M$ admit Hermitian-Yang-Mills metric then we have the following known universal inequality of Yau-Uhlenbeck $$\int_Mc_2(End E)\wedge \...
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5 votes
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Darboux's Theorem

I read proofs of Darboux's theorem that a symplectic form $\omega$ on M locally is symplectomorphic to a standard symplectic form $\omega_{0}= \sum dp \wedge dq$ in Rolf Berndt's An Introduction to ...
Dai's user avatar
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Fibered 3-manifolds over circle with harmonic projection map

Suppose we have a surface bundle $M$ over the unit circle $S^1$ and $M$ is assigned with Riemannian metric $g$. The projection map $\pi: M\to S^1$ now can be homotopic to a harmonic map $\phi: M\to S^...
Donghao's user avatar
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Understanding projective space as fibrations of tori over spaces with boundaries

The toric manifold $\mathbb{CP}^1$ can be understood as a circle fibration over an interval $I$, with the circles having zero radius at the boundaries of the interval. How does one generalize this ...
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Higher-dimensional version of Fenchel's theorem that total curvature is $\ge 2 \pi$

Q. Is there a higher-dimensional version of the theorem due to Fenchel that the total curvature of a closed curve in $\mathbb{R}^3$ is $\ge 2\pi$, with equality only if the curve is planar and convex? ...
Joseph O'Rourke's user avatar
2 votes
1 answer
2k views

Mean curvature vector approximated for the discrete Laplace Beltrami Operator

"It is known that, for a curvature continuous surface, if the curve $\gamma$ shrinks to the point $v_i$, the (line) integral converges to the mean curvature $\kappa(v_i)$ of the surface at the point $...
Joei's user avatar
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Interplay between Algebraic and Differential Geometry

Apologies to start with if this question is 'too soft' or not really research level. I'm a graduate student interested in both algebraic and differential geometry, although very much a novice with the ...
A. Thomas Yerger's user avatar
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3 answers
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What results are immediately generalised to higher dimensions, in light of Schoen and Yau's recent preprint?

Many problems in geometric analysis and general relativity have been established in dimensions $3\leq n\leq 7$, as the regularity theory for minimal hypersurfaces holds up to dimension 7*. In a recent ...
Steve McCormick's user avatar
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What is the definition of "geometric analysis"? [closed]

Recently it has been brought to my attention that the subject "geometric analysis" is not even well-defined (unlike the subject partial differential equations, algebraic geometry, etc). Can someone ...
Zhexiu Tu's user avatar
7 votes
1 answer
641 views

Negatively curved manifolds with many totally geodesic submanifolds

I'm curious about the following question and have not been able to find any literature on the topic: Suppose that $M$ is a closed negatively curved Riemannian manifold with a "large" quantity of ...
Clark's user avatar
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1 answer
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Zeros of polynomials as a finite union of manifolds

Consider a polynomial in $d$ variables, $p:\mathbb{R}^{d}\rightarrow\mathbb{R}$. Denote by $\mathcal{C}$ its set of zeros, i.e. $$\mathcal{C}=\{x\in\mathbb{R}^{d}\ |\ p(x)=0\}.$$ Q. Is it possible ...
Arturo's user avatar
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Area of a sphere in Alexandrov spaces

Let $(X,d)$ be an $n (\geq2)$ dim Alexandrov space with curvature $\geq k$. $B(x,r)$ is an open ball in $X$. Let $M_{k,n}$ be the $n$ dim space form of constant curvature $k$. $B_k(r)$ is an open ball ...
user84068's user avatar
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Volume of $SL(2,\mathbb{C})$ [closed]

So according to http://www-users.math.umn.edu/~garrett/m/v/SL2C.pdf I can write the Haar measure of $SL(2,\mathbb{C})$ as $$d\mu = \sinh^2(r) dr dk dk'$$ where $r$ runs over nonnegative real numbers ...
Alireza Behtash's user avatar
4 votes
1 answer
136 views

Convex hull of a connected subset on a complete surface of non-positive curvature

Let $S$ be a simply connected surface, possibly with boundary components, with a smooth complete metric of non-positive curvature. Let $X\subset S$ be a closed connected subset. I would like to know ...
aglearner's user avatar
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3 votes
1 answer
188 views

Constructing a Hamiltonian $S^1$-action on a neighborhood of a symplectic divisor

Let $M^{2n}$ be a symplectic manifold and let $M^{2n-2}$ be a symplectic submanifold. How to construct a non-trivial Hamiltonian $S^1$-action on $M^{2n-2}$ on a small neighborhood of $M^{2n-2}$, that ...
aglearner's user avatar
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1 vote
1 answer
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Polar coordinates of a set with different radius and angle

Let $M$ be a $2$-dimensional Riemannian manifold and let $U\subset M$ be an open set. Suppose there exist polar coordinates $(r,\theta)$ with center $q\in M$ such that $$U=\lbrace{ (r,\theta): 0<...
Sammyy Delbrin's user avatar
6 votes
2 answers
358 views

Sources for Alexandrov surfaces

There are two distinct notions in differential geometry associated with A. D. Alexandrov: (1) Alexandrov spaces of courvature bounded from below; (2) Alexandrov surfaces of bounded total curvature (...
Mikhail Katz's user avatar
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3 votes
3 answers
833 views

Good exposition of "Calabi ansatz"

As far as I understand, Calabi ansatz is (in particular) a way to produce Kähler metrics on total spaces of line bundles (or their disk subbudles) over Kähler manifolds of the following form: Calabi ...
aglearner's user avatar
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Proof of the holomorphic Frobenius theorem in Voisin's book on Hodge theory (Theorem 2.26)

I'm trying to understand the proof of the holomorphic version of the Frobenius integrability theorem given in p. 51-52 of Voisin's text "Hodge Theory and Complex Algebraic Geometry I". Statement: ...
Saal Hardali's user avatar
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On the embedding of manifolds into infinite-dimensional spaces

Let $X$ be a (connected, finitely dimensional) topological/smooth/complex manifold and let $i$ be a weakly continuous/continuous/smooth/holomorphic map from $X$ into the dual $F^{*}$ of a real or ...
erz's user avatar
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12 votes
1 answer
383 views

Spin structures on Sasakian manifolds and the Kähler analogy

A Sasakian manifold is often said to be the odd dimensional analogue of a Kähler manifold. Now for a $2n$-dimensional Kähler manifold we know from Atiyah that it is spin exactly if the line bundle $\...
Alesandro Levi's user avatar
2 votes
0 answers
247 views

Does any non-Hausdorff manifold admit a metric tensor of signature $(p,q)$?

Hicks in his book, "Notes on differential geometry", works with manifolds by specifically stating that they are not required to be Hausdorff unless otherwise stated. He then goes on to define the ...
Slereah's user avatar
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919 views

Complexification of smooth manifolds

Given a $C^\infty$- smooth manifold $M$, is there a natural way to complexify it? By this I mean finding a complex manifold $N$ and a smooth function $f:M\to N$ such that $df:T_xM\otimes \mathbb{C}\to ...
Omar's user avatar
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0 answers
240 views

Exponential map for non-smooth Finsler manifolds

Context I'm interested in studying reversible Finsler manifolds which do not have the strong convexity of the Hessian property (that is the Finsler function is a regular norm on every tangent space). ...
ABIM's user avatar
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Geometric meaning of Aomoto complex

Generally, if we have any space $X$, then multiplication by any odd class $\eta$ makes $H^*(X)$ a complex (usually called Aomoto complex) $A_{\eta}$ because cup product is graded commutative. It is ...
Denis T's user avatar
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1 answer
86 views

The sheaf propagation is open in the zero section

Let $X$ a smooth manifold and $F$ a sheaf (let's say of abelian groups) on $X$. We will say that $F$ propagates at $x\in X$ in the (co)-direction $p \in T_x^*X$ if for all $C^1$-function $\phi$ ...
C. Dubussy's user avatar
2 votes
0 answers
280 views

"Riemannian" collar theorem

Let $(M,g)$ be a compact manifold of dimension $n$ with boundary. If $\partial M$ is smooth then one has a control on the determinant of the Jacobian of the diffeomorphism in the collar theorem, i.e....
Math101's user avatar
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19 votes
7 answers
2k views

Supermanifolds — elementary introduction?

I am looking for an elementary but mathematically precise introductory text on supermanifolds in a modern differential geometric setting. Elementary in the sense that there is plenty of motivation for ...
Arnold Neumaier's user avatar
4 votes
1 answer
983 views

Smoothness of the square of the distance function on a Riemannian manifold

Let $(M^n,g)$ be a smooth Riemannian manifold. The distance between two points is the infimum of the lengths of the curves which join the points. Consider the square of the distance function $d^2\...
MatBoss918's user avatar
5 votes
0 answers
184 views

Length and average height of Gerver's Sofa

I am currently investigating possible expansions and generalizations of the moving sofa problem. I am mainly considering higher dimensions and other angles. This led me to consider the height and ...
user avatar
11 votes
1 answer
2k views

Relationship between the Witt algebra and vector fields on the circle

I have seen some (apparently contradictory) claims by mathematicians and physicists regarding the existence of an (infinite dimensional) Lie group whose Lie algebra is the Virasoro algebra. The ...
pre-kidney's user avatar
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13 votes
0 answers
354 views

Euler characteristic of *hyperbolic* orbifolds

This is a follow-up to this question. Which rational numbers arise as Euler characteristics of orbifold quotients of $\mathbb{H}^n?$ The answer is not even clear for $n=2.$ It is clear that the Euler ...
Igor Rivin's user avatar
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16 votes
3 answers
1k views

SO(3) action on (simply connected) 6 manifold with discrete fixed point

If a 6-dimensional orientable smooth manifold $M$ admits a smooth effective $SO(3)$ action with discrete fixed point set, what can we say about the topology of $M$? What if we assume that in addition ...
Yuhang Liu's user avatar
4 votes
0 answers
150 views

Trivialization of a fibration after a base change

Let $E\to B$ be a locally trivial fibration of curves of genus >1 (everyhting is compact and over $\mathbb{C}$). I read that this fibration becomes trivial "after a finite unramified base change". ...
Andrew's user avatar
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4 votes
0 answers
157 views

Analytic maps $\varphi: \mathbb C^n\to \mathbb C^n$ with degenerate differentials

Let $B^n\subset \mathbb C^n$ be a unit ball with center $p$ . Let $\varphi: B^n\to \mathbb C^n$ be a complex analytic map such that $d\varphi$ has rank at most $n-1$ at $p$. I would like to know if ...
aglearner's user avatar
  • 14k
8 votes
1 answer
569 views

Smoothing of a Kähler orbifold metric on a complex surface

Let $S$ be a smooth complex projective surface and $D\subset S$ be a smooth complex curve. Fix an integer $m>1$ and consider $(S,D,m)$ as an orbifold with orbi-locus $D$ with stabilizer $\mathbb ...
aglearner's user avatar
  • 14k
5 votes
1 answer
397 views

Fundamental group of compact manifolds with non-negative Ricci curvature and Bieberbach theorem

Let $M$ be a compact n-dimensional Riemannian manifold with non-negative Ricci curvature. Then its universal cover $\tilde{M}$ is isometric to $\mathbb{R}^p\times N$ for some $p\leqslant n$ and $N$ ...
mathmetricgeometry's user avatar
23 votes
2 answers
1k views

Is every rational realized as the Euler characteristic of some manifold or orbifold?

Let me first ask the question for two-dimensional compact, connected manifolds and orbifolds. Then, if the answer is No, one can remove various conditions on the dimension, and allow non-compact ...
Joseph O'Rourke's user avatar
9 votes
1 answer
472 views

What does positivity of the first Pontryagin number of a vector bundle tell us?

Some context: In the theory of compact, oriented Riemannian Einstein 4-manifolds, there is a a fundamental topological constraint that is implied by the Einstein equations. To wit, if $\chi$ and $\...
Brian Klatt's user avatar
15 votes
2 answers
2k views

Every 4-manifold has a $\operatorname{Spin}^c$ Structure

$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SO{SO}$I'm having trouble understanding the proof given in Morgan's The Seiberg–Witten Equations and Applications to the Topology of Smooth Four-...
jdk3264's user avatar
  • 151
9 votes
1 answer
730 views

On the topology induced by a Lorentzian metric

Let $(M,g)$ be a time-oriented smooth Lorentzian manifold, with Lorentzian metric $g$. In the following thread: https://physics.stackexchange.com/questions/228669/why-pseudo-riemannian-metric-cannot-...
Bilateral's user avatar
  • 3,064
44 votes
2 answers
3k views

Bijection from the plane to itself that sends circles to squares

Let me apologize in advance as this is possibly an extremely stupid question: can one prove or disprove the existence of a bijection from the plane to itself, such that the image of any circle ...
Tom Solberg's user avatar
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1 vote
0 answers
102 views

Different conventions for Hirzebruch-Riemann-Roch?

I seem to find a disparity by a factor of 2 in some results where the Hirzebruch-Riemann-Roch theorem is used. I am particularly troubled by the following example; in https://arxiv.org/abs/0707.2786 (...
Mtheorist's user avatar
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2 votes
0 answers
232 views

What exactly is the role of the mysterious manifold underlying the definition of a superspace?

In the intro to chapter 12.3 of this book about the applications of coherent states, it says that classical spaces for bosons are real or complex vector spaces or manifolds, whereas classical spaces ...
Dilaton's user avatar
  • 408
2 votes
1 answer
181 views

Relation of pseudo-torsion with curvature in degenerate plane

Question: I'd like to know if there is some reference or reasonable way to develop curve theory in a plane with degenerate metric $(\Bbb R^2, {\rm d}s^2 ={\rm d}x^2)$. Context: In Lorentz-Minkowski ...
Ivo Terek's user avatar
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8 votes
2 answers
411 views

On the causal structure of spacetimes: piecewise $C^1$, $C^k$ or $C^\infty$?

This is a more technical question but it seems that there is some confusion in the literature on the choice of curves used to define the causal relations in time-oriented Lorentz manifolds: the ...
Stefan Waldmann's user avatar
3 votes
1 answer
188 views

Index of linearized operator for symplectic vortex equations

In reference [1], the index of the linearized operator for the symplectic vortex equations is computed on page 27-28. The first step of the proof says that the operator \begin{equation}\tag{1} \...
Mtheorist's user avatar
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