Questions tagged [dg.differential-geometry]

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

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Conformal laplacian on asymptotically flat manifolds with boundary

Let $g$ be an asymptotically flat metric on $M = \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball. Suppose $X$ is a smooth vector field on $M$ that is decaying exponentially and satisfies $$\...
Laithy's user avatar
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Lie group framing and framed bordism

What is the definition of Lie group framing, in simple terms? Is the Lie group framing of spheres a particular type of Lie group framing? (How special is the Lie group framing of spheres differed ...
zeta's user avatar
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2 votes
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The existence of a positive Green function for the Laplacian on $\mathbb R$

One can show explicitly and easily that the function $G(x,y) = \frac 1 2 |x-y|$ is a positive Green function for the Laplacian $\frac {\mathrm d ^2} {\mathrm d x ^2}$ on $\mathbb R$ (endowed with the ...
Alex M.'s user avatar
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4 votes
1 answer
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Bounded covariant derivative of curvature tensor

Let $M$ be a complete Riemannian manifold. Suppose that there are positive constants $i_0$ and $K$ such that the injectivity radius of $M$ is at least $i_0$ and $|\mathrm{Rm}|\le K$ and $|\nabla \...
Anton Petrunin's user avatar
1 vote
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Minimal first Pontryagin class $p_1=1$?

From Hirzbuch theorem, the signature of 4-manifold $\sigma = p_1/3$ with the first Pontryagin class $p_1$. I know that the $\sigma=p_1/3 =1$ so $p_1=3$ for complex projective space $\mathbb{P}^2$. Is ...
zeta's user avatar
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How do we calculate the gradient of this function defined using the Riemannian logarithm on a Riemannian manifold?

We consider the following function $\psi$ on an open subset $V\subset M,$ a Riemannian manifold of dimension $m,$ so that $\exp_p:U\to V$ is a diffeomorphism with its inverse $\log_p: V\to U$. Let $v\...
Learning math's user avatar
4 votes
1 answer
163 views

Convex hull of 3 points in Cartan-Hadamard manifolds

Can the convex hull of $3$ points in a Cartan-Hadamard manifold be smooth? A Cartan-Hadamard manifold $M$ is a complete simply connected manifold with nonpositive curvature (so it includes the ...
Mohammad Ghomi's user avatar
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2 answers
235 views

Convergence of metric spaces of increasing dimension

Given two metric spaces we can define the Gromov-Hausdorff (GH) distance. There are compactness results stating that a sequence of manifolds of a fixed dimension, with a uniform lower Ricci bound and ...
theflame's user avatar
2 votes
1 answer
142 views

string bordism group and framed bordism group for $d \leq 6$ and $d \geq 7$

Why do the string bordism group and the framed bordism group coincide the same in dimensions lower than 7 ($d = 0,1,2,3,4,5, 6$)? Why do the string bordism group and the framed bordism group differ ...
wonderich's user avatar
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Derive distributional inequalities from pointwise estimates

My question is how to prove the following claim: Suppose that $E$ is an algebraic set in $\mathbb{R}^n (n\ge3)$ with dimension $\le n-2$, and $u$ is locally Lipschitz continuous on $\mathbb{R}^n$. If ...
William's user avatar
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Decompositions of $\partial_i$ to the radial direction and rotations in higher dimensions

We know in dimension $3$, \begin{align} \partial_{i}= \frac{x_i}{r} \partial_{r} - \varepsilon_{ijk} \frac{x^j}{r} \frac{R^k}{r} , \end{align} where $\varepsilon_{ijk}$ are Levi-Civita symbols ...
lsb's user avatar
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Gauge Lie groupoid associated to $SO(3)$ double cover

From each Lie group $G$ and principal $G$-bundle $P \rightarrow E$ one can form an associated (or gauge) Lie groupoid as the quotient of pair groupoid by the action of $G$ on $P \times P$ $$ \frac{P \...
Alexander Golys's user avatar
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1 answer
146 views

Examples of hyperbolic manifolds of dimension $\geq$ 3 with disjoint totally geodesic hypersurfaces

I am hoping to find examples of compact hyperbolic manifolds with at least 2 disjoint totally geodesic hypersurfaces. Ideally, I would like examples in dimension at least 4, though 3-dimensional ...
ಠ_ಠ's user avatar
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3 votes
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Critical points up to smooth homotopy

Let $M$ and $N $ be closed connected smooth manifolds of dimension $n$. Let $f: M\rightarrow N$ be a smooth function and not null-homotopic. Is there a smooth homotopy $H: [0,1]\times M\rightarrow N$ ...
GSM's user avatar
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Is any globally-hyperbolic manifold conformally equivalent to one with complete slices?

Let $(M,g)$ be a globally-hyperbolic Lorentzian manifold. By the seminal work of Geroch and Bernal-Sánchez, we know that $$M=\mathbb{R}\times\Sigma,\,\,\,\quad g=-\beta^{2}dt^{2}+h_{t}$$ where $\Sigma$...
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Curve length in the Sasaki metric

I am trying to read Appendix II.A.2 (Distances in the tangent bundle) in Canary, Epstein, Marden (eds.), Fundamentals of Hyperbolic Manifolds: Selected Expositions and am stumbling over a calculation ...
Jochen Trumpf's user avatar
3 votes
1 answer
306 views

reference for reading Schoen Yau positive mass theorem proof II

I am trying to read the paper by Schoen and Yau, Proof of the Positive Mass Theorem II. The notation is very different from what I am familiar with (basically Robert Wald's book on general relativity)....
Bowen Zhao's user avatar
7 votes
0 answers
674 views

What's the point of geometric representation theory?

Please forgive the provocative title, what I mean is the following: One can find representations of Lie algebras in geometric settings, the most famous being the Bott–Borel–Weil theory. However, ...
Béla Fürdőház 's user avatar
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Renormalized limit measure on a splitting Ricci limit space

I'm reading Cheeger&Colding's On the structure of spaces with Ricci curvature bounded below. I recently. In the proof of proposition 1.35 as follows. I got some questions: Proposition 1.35. Let $\...
eulershi's user avatar
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What is known about warped product metrics satisfying conditions more general than conformal flatness?

In this paper, the authors characterize warped product metrics which are conformally flat (the fibers must have constant sectional curvature, on some cases there is a limitation on the number of ...
Matheus Andrade's user avatar
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77 views

Tangent spaces of Lipschitz sub manifolds

Consider $\mathbb{R}^n$, $k<n$, and topological embeddings (homeomorphisms onto image) $f_i : \mathbb{R}^k \supseteq B_1(0) \to \mathbb{R}^n$, $i=1,2$, which are also Lipschitz continuous and ...
jsb's user avatar
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different definitions of holomorphic bisectional curvature

Peter Li and Jiaping Wang defined holomorphic bisectional curvature in their paper as follows: Assume that $M^m$ is a Kahler manifold of complex dimension $m$. Let $ \{e_1, \cdots , e_m\} $ be a ...
HeroZhang001's user avatar
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0 answers
47 views

Rotational invariance of Laplace-Beltrami eigenvalue problem on smooth manifolds

I am currently looking at the eigenvalue problems of the Laplace-Beltrami operator. Let $(M,g$) be a smooth and oriented Riemann manifold. I am investigating the eigenvalue problem of the Laplace-...
SebastianP's user avatar
3 votes
0 answers
59 views

Covariant derivative $\nabla_{\dot{\gamma}} \dot{\gamma}$ of constrained velocity vector $\dot{\gamma}$ by distribution $B$ and bounded map $\delta_i$

I have been studying differential geometry for a while. The subject is hard to grasp at first, but gets easier once one understands the main concepts. One of them are covariant derivative $\nabla_X Y$....
Usuário 6789's user avatar
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48 views

Sufficient conditions for chain recurrent set equal to set of non wandering points

Given a generic diffeomorphism, I know that the set of nonwandering points is contained in the chain recurrent set, but the converse is not always true. Is there some sufficient conditions under which ...
giangian's user avatar
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91 views

Asymmetric minimal surfaces in $H^3$

Inspired by this question, are there any explicit parameterizations of asymmetric minimal surfaces in $\mathbb{H}^3$? E.g. something like the minimal surface which lies over the ellipse given by $$y^2 ...
JMK's user avatar
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106 views

Changing the sign of the moment map in the Seiberg Witten equations

The Seiberg-Witten equations on a closed four manifold $$ D_A \varphi = 0, F_A^+ = \mu(\varphi) $$ are elliptic (up to gauge transformations), and so the equations $$ D_A \varphi = 0, F_A^+ = -\mu(\...
user2271513's user avatar
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$C^1$ manifold with complex structure

Let $M$ be a manifold. A complex structure on $M$ is an endomorphism $J \in \text{End}(TM)$ such that $J^2 = -\text{id}$ together with the vanishing of the Nijenhuis tensor. If $J$ is real-analytic, ...
Chicken feed's user avatar
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1 answer
144 views

Going from piecewise to genuine geodesic without decreasing number of intersections?

Let $(M^2,g)$ be a complete, two-dimensional Riemannian manifold be given; also given is $\gamma: [0,\infty) \to M$, an injective geodesic in $M$. Suppose there are two geodesic segments $\gamma_i : [...
Leo Moos's user avatar
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16 votes
0 answers
390 views

Is the oriented bordism ring generated by homogeneous spaces?

I am trying to find a Riemannian geometrically well-understood set of generators of the oriented bordism ring, including the torsion parts. By a set of generators, I mean that the set generates the ...
Zhenhua Liu's user avatar
12 votes
3 answers
946 views

Area of a smooth complex projective curve

Let $P(X,Y,Z)$ denote a homogeneous polynomial in $\mathbb{C}[X,Y,Z]$ such that $X_P = \{(u : v : w) \in \mathbb{C}\mathbb{P}^2 \mid P(u,v,w) = 0\}$ defines a smooth complex projective curve in $\...
Daniel Asimov's user avatar
1 vote
1 answer
335 views

Topological degree of differentiable map using line integrals?

Let $f:\mathbb R^2 \to \mathbb C$ be a $C^1$ function that vanishes at a point $x_0.$ I can then define $$-i \int_{\gamma_\varepsilon} \nabla \log(f(s)) \cdot ds := - i \int_0^1 \nabla (\log f)(\...
António Borges Santos's user avatar
1 vote
0 answers
101 views

Planar sections of convex sets in Cartan-Hadamard manifolds

Let $X$ be a convex set in Euclidean space $\mathbf{R}^n$ and $p\in\mathbf{R}^n$ be a fixed point. Then any plane $\Pi$ passing through $p$ intersects $X$ in a convex set. Conversely, this property ...
Mohammad Ghomi's user avatar
3 votes
1 answer
337 views

Does Hermite-Einstein imply Kähler-Einstein?

Let $M$ be a compact Kähler manifold and let $\nabla$ be its Levi-Civita, or equivalently its Chern, connection. Denoting the vector bundle of complexified one forms of $M$ by $\Omega^1_{\mathbb{C}}$, ...
Didier de Montblazon's user avatar
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0 answers
116 views

Dirac distribution on a manifold $M$ as a smooth manifold in $C^1(M)^*$, question about its dimension

I have not learned many knowledges on differential geometry, I met this when trying to read the min-max scheme in PDE on manifold, which is in Section3.1. Let $M_1= \delta_{x_i}$, $x_i \in M$. For $\|...
Elio Li's user avatar
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3 votes
1 answer
228 views

1D topological defects in $d>3$ spatial dimensions

I am trying to construct a 1D topological defect solution in 4 spatial dimensions, i.e., a solution to some PDE (likely the equations of motion of some Lagrangian) on $\mathbb{R}^{4}$ which is ...
math_lover's user avatar
7 votes
1 answer
462 views

The tangent map of the exponential map

Let $(M,g)$ be a Riemannian manifold and let $\exp^M:TM\to M$ denote the exponential map. Its tangent map $T\exp^M$ is a map $TTM\to TM$. The connection on $TM$ induces a canonical metric on $TM$, the ...
Raz Kupferman's user avatar
5 votes
0 answers
334 views

Injectivity of div–curl operator

$\DeclareMathOperator\div{div}\DeclareMathOperator\curl{curl}$Consider a div–curl system \begin{align*} Lu &= (\div(u), \curl(u)) \text{ in } \Omega \subset M, \text{ a 3-manifold}, \\ u &= 0 \...
Chris's user avatar
  • 389
14 votes
2 answers
2k views

Is a manifold Euclidean if its tangent bundle is Euclidean?

I'm wondering whether an $n$-dimensional manifold diffeomorphic to $\mathbb{R}^n$ if its tangent bundle is diffeomorphic to $\mathbb{R}^{2n}$. Many thanks!
Felix's user avatar
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5 votes
1 answer
627 views

Ricci flow + Nash embedding

I have a basic question about how geometric flows such as the Ricci flow interact with the Nash embedding theorem. Say you have a 1-parameter family of Riemann metrics on a compact manifold $N$. If ...
Ryan Budney's user avatar
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1 vote
0 answers
303 views

About "residual" scalar curvature in Einstein warped product manifold

I have an Einstein warped product manifold $M=B \times_f F$ with warped metric $g=g^B+f^2g^F$, where $F$ is Ricci flat, then its scalar curvature is $S^F=0$. It is well known that the scalar curvature ...
MathDG's user avatar
  • 242
18 votes
1 answer
2k views

Intuition behind manifolds which are homeomorphic but not diffeomorphic

Popular articles on mathematics often explain the difference between homeomorphism and diffeomorphism with statements like - "A rectangle is homeomorphic to the circle but not diffeomorphic to it&...
Anindya's user avatar
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3 votes
0 answers
67 views

Is it always possible to find a conjugate optical function?

Optical functions (functions with null gradients) and double null foliations (foliations of a spacetime by two related optical functions) play a large roll in modern mathematical relativity research. ...
Chris's user avatar
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1 vote
0 answers
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Lifting action of torus to torus bundle

Preamble: Let $X$ be a simply connected smooth manifold and $P \to X$ be a principal $T^\ell$ bundle on it. Let $\phi$ be a smooth action of $T^k$ on $X$. The paper "Lifting compact group actions ...
Nicolò Cavalleri's user avatar
1 vote
0 answers
40 views

Verdier (w) condition implies the $w_f$ condition when the restriction of $f$ in each stratum is a submersion?

Let $X\subset\mathbb{R}^n$ be and let $\Theta=(X_\beta)_{\beta\in I}$ a Verdier stratification for X. Let $f:X\rightarrow\mathbb{R}$ be a polynomial function, such that $f_{|_{X_\beta}}$ is submersion ...
Giovanny Snaider Barrera Ramos's user avatar
0 votes
0 answers
113 views

Sufficient condition for existence of a closest-point projection from a neighborhood onto a subset of a Riemannian manifold

Let $(M,g)$ be a Riemannian manifold and let $N$ be a subset of $M$. On one hand, it is well known that if $N$ is an embedded submanifold of $M$, then it admits a tubular neighborhood, and, ...
gpr1's user avatar
  • 134
1 vote
0 answers
52 views

Limits of branched minimal immersions into the sphere

Can a sequence of branched minimal immersions $M_j^n$ in the round sphere $S^{n+1}$ converge to a smoothly embedded $\Sigma$, in the sense that $ M_j \to 2 \Sigma$ as currents or varifolds? The case ...
Leo Moos's user avatar
  • 4,968
2 votes
0 answers
136 views

Hypercomplex structures and tangent space decompositions

For any almost complex manifold we have a decomposition of its tangent space into two subspaces $T = T^{(1,0)} \oplus T^{(0,1)}$. For an almost hypercomplex manifold we have three almost-complex ...
Diego de la Paz's user avatar
1 vote
0 answers
43 views

Can we bound the squared Gaussian curvature of genus three triply periodic minimal surfaces?

Assume that $\mathcal{M}$ is a balanced triply periodic minimal surface of genus 3, embedded in a flat torus $T^3=\mathbb{R}^3/\Lambda$ for a lattice $\Lambda$ with volume 1. I want to understand the ...
Matthias Himmelmann's user avatar
2 votes
0 answers
104 views

A specific question in $G_2$ geometry

Let's take a closed $G_2$ manifold $(M,\Phi)$. $\Phi$ denoting the three-form which defines the $G_2$ structure on $M$. Let's take a closed two form $\theta\in\Omega^2(M).$ Is \begin{align} d(\theta_{\...
Partha's user avatar
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