Questions tagged [dg.differential-geometry]
Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.
8,645
questions
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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 \...
7
votes
1
answer
330
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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_{...
3
votes
1
answer
507
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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....
6
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1
answer
481
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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 \...
5
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0
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528
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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 ...
5
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235
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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^...
2
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190
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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 ...
1
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293
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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?
...
2
votes
1
answer
2k
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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 $...
4
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0
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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 ...
52
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3
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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 ...
4
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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 ...
7
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1
answer
641
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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 ...
5
votes
1
answer
227
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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\}.$$
Q. Is it possible ...
3
votes
1
answer
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Area of a sphere in Alexandrov spaces
Let $(X,d)$ be an $n (\geq2)$ dim Alexandrov space with curvature $\geq k$. $B(x,r)$ is an open ball in $X$. Let $M_{k,n}$ be the $n$ dim space form of constant curvature $k$. $B_k(r)$ is an open ball ...
2
votes
0
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590
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Volume of $SL(2,\mathbb{C})$ [closed]
So according to http://www-users.math.umn.edu/~garrett/m/v/SL2C.pdf
I can write the Haar measure of $SL(2,\mathbb{C})$ as
$$d\mu = \sinh^2(r) dr dk dk'$$
where $r$ runs over nonnegative real numbers ...
4
votes
1
answer
136
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Convex hull of a connected subset on a complete surface of non-positive curvature
Let $S$ be a simply connected surface, possibly with boundary components, with a smooth complete metric of non-positive curvature. Let $X\subset S$ be a closed connected subset. I would like to know ...
3
votes
1
answer
188
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Constructing a Hamiltonian $S^1$-action on a neighborhood of a symplectic divisor
Let $M^{2n}$ be a symplectic manifold and let $M^{2n-2}$ be a symplectic submanifold. How to construct a non-trivial Hamiltonian $S^1$-action on $M^{2n-2}$ on a small neighborhood of $M^{2n-2}$, that ...
1
vote
1
answer
98
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Polar coordinates of a set with different radius and angle
Let $M$ be a $2$-dimensional Riemannian manifold and let $U\subset M$ be an open set. Suppose there exist polar coordinates $(r,\theta)$ with center $q\in M$ such that
$$U=\lbrace{ (r,\theta): 0<...
6
votes
2
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358
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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 (...
3
votes
3
answers
833
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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 ...
6
votes
2
answers
1k
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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: ...
5
votes
0
answers
103
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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 ...
12
votes
1
answer
383
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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 $\...
2
votes
0
answers
247
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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 ...
9
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0
answers
919
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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 ...
3
votes
0
answers
240
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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). ...
8
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0
answers
273
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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 ...
3
votes
1
answer
86
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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 $\phi$ ...
2
votes
0
answers
280
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"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....
19
votes
7
answers
2k
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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 ...
4
votes
1
answer
983
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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\...
5
votes
0
answers
184
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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 ...
11
votes
1
answer
2k
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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 ...
13
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0
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354
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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 ...
16
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3
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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 ...
4
votes
0
answers
150
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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". ...
4
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0
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157
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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 ...
8
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1
answer
569
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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 ...
5
votes
1
answer
397
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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 $N$ ...
23
votes
2
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1k
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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 ...
9
votes
1
answer
472
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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 $\...
15
votes
2
answers
2k
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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-...
9
votes
1
answer
730
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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-...
44
votes
2
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3k
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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 ...
1
vote
0
answers
102
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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 (...
2
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232
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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 ...
2
votes
1
answer
181
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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 ...
8
votes
2
answers
411
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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 ...
3
votes
1
answer
188
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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}
\...