Questions tagged [dg.differential-geometry]

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

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A geometric criterion for uniqueness in the Plateau problem?

Let $\gamma: S^1 \to \partial B \subset \mathbf{R}^3$ be a smooth, simple closed curve in the boundary of the unit ball. Suppose that $\gamma$ intersects every horizontal plane $\Pi_t = \{ z = t\}$ at ...
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6 votes
2 answers
489 views

Consequences of Nash-Tognoli Theorem

The Nash-Tognoli theorem states that every closed and smooth manifold is diffeomorphic to a real algebraic variety. This appears to me as a very strong and surprising fact. However, I am not aware of ...
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3 votes
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47 views

A strong maximum principle for varifolds of arbitrary codimension

Let $M$ be an $n$-dimensional Riemannian manifold and $N$ a hypersurface in $M$. Let $p \in N$ and $\kappa_1 \leq \cdots \leq \kappa_{n-1}$ be the principal curvatures of $N$ at $p$ with respect to a ...
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Examples of Geometrically meaningful constraint to component of derivative similar to Cauchy Riemann equations

In most of basic Complex analysis, the geometric depth of the subject arises from the PDE that the component of derivative must obey for the function to be considered complex differentiable (/Cauchy ...
3 votes
0 answers
45 views

What are known properties of the boundary curves of J-holomorphic curve with boundary

Suppose $\Sigma$ be a punctured Riemann surface with punctured boundary, and $(M, J)$ be a $2n$-manifold with almost complex structure $J$. Let $L$ be a totally real submanifolds of $M$ in smooth ...
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8 votes
1 answer
149 views

Convex hulls of compact sets in a 2-manifold

Let $(\mathbb{R}^2,g)$ be a complete Riemannian manifold. Let $K\subset \mathbb{R}^2$ be a compact, connected set, and let $\text{conv}(K)$ be its convex hull, i.e., the intersection of all ...
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0 votes
0 answers
73 views

Cohomology in a combinatorial way using ribbon graphs

I am interested in studying the cohomology of surfaces. Let $S$ be a compact orientable connected surface. One possible way is to learn cohomology using differential forms. Is it possible to approach ...
2 votes
0 answers
64 views

Jacobi equation and conjugate points on solution curves of the Van der Pol vector field

Let $X$ be a geodesible non vanishing vector field on a manifold $M$. Namely there is a Riemannian structure $(M,g)$ such that all integral curves of $M$ are unparametrized geodesics of the ...
18 votes
2 answers
811 views

Statements in differential geometry independent from ZFC

It is well known that some problems in functional analysis and in general topology are independent from ZFC: to name a few, Kaplansky's conjecture, the existence of outer automorphisms of the Calkin ...
2 votes
0 answers
66 views

Generalizations of elliptic chain complexes

I would like to know if it is possible to generalize the notion of elliptic chain complex of differential operators to different contexts, whether geometric or non geometric. I have in mind D-geometry....
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5 votes
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Intersections of geodesics in an "almost flat" plane

Let $g$ be a complete metric on $\mathbb{R}^2$, such that: Outside of a compact connected set $K\subset \mathbb{R}^2$, the curvature of $g$ vanishes. The integral of the Gaussian curvature in $K$ is ...
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2 votes
1 answer
154 views

Classification of Lie group structures on $\mathbb{R}^n$

Is it possible to describe, up to isomorphism, all Lie groups $G$ whose underlying manifold is diffeomorphic to $\mathbb{R}^n$ (with its standard smooth structure)? In fact, I haven't found any such ...
1 vote
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157 views

Monodromy group action on de Rham cohomology

Let $f : Y \longrightarrow X := \mathbb{P}^1\setminus\{0,1,\infty\}$ be the smooth proper morphism associated to the Legendre family, which is an elliptic fibration of the punctured line, with fibre ...
2 votes
1 answer
70 views

On equipartitions of surfaces of 3D convex regions

Let S be the surface of a 3D convex region (a 'convex surface'). Let S' be a subset of S. We shall refer to S' as geodetically convex in S if the following condition holds: If A and B are two points ...
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5 votes
1 answer
331 views

Six people standing on earth

Consider 6 people $p_i$, $i=1,\dots 6$, standing on a sphere $S^2$. We label the positions of these people by $p_i$ again. Suppose no pair of these points $p_i$ are antipodal. At each point $p_i$ ...
5 votes
0 answers
105 views

Intersection of orbits of earthquake flow on Teichmüller space

Let $\Sigma$ be a closed oriented surface of genus $g\geq2$. We consider $\mu$ and $\nu$ two filling measured laminations on $\Sigma$. (We say that $\mu$ and $\nu$ fill $\Sigma$ if $\Sigma\setminus(\...
2 votes
0 answers
40 views

Riemannian submanifolds of $2$-Wasserstein space

In the article "Wasserstein Geometry Of Gaussian Measures" by Asuka Takatsu the author shows how the space of d-dimensional Gaussian probability measures with non-singular covariance ...
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4 votes
1 answer
129 views

Vanishing directional derivatives on $S^2$

Let $u$ be a smooth function defined on the unit sphere $S^2$. Does there exist a plane $P$ passing through the origin such that $P\cap S^2$ contains at least three points $x_1,x_2,x_3$ with $\nabla u(...
1 vote
1 answer
148 views

Decomposition of tensor field on hypersurface

Let $(\mathcal{M},g)$ be a Lorentzian manifold, which is globally of the form $\mathcal{M}\cong I\times\Sigma$, where $I\subset\mathbb{R}$ ("time") and $\Sigma$ ("space") is some $...
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8 votes
1 answer
134 views

Is the $n/2$-th heat kernel coefficient topological?

I have asked the same question on math.SE, without much success so I'm trying my luck here too. Let $M$ be an $n$-dimensional manifold, with $n$ even and consider the heat kernel of the Laplacian on $...
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1 vote
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63 views

Is Stenzel's Ricci-flat metric on $T^*\mathbb{CP}^n$ hyperkahler?

In a well-known paper, Stenzel constructed complete Ricci-flat Kahler metrics on the total spaces of cotangent bundles of $S^n$, $\mathbb{RP}^n,$ $\mathbb{CP}^n$, $\mathbb{HP}^n$, and $\mathbb{OP}^2$. ...
5 votes
1 answer
196 views

Closed geodesics on bumpy spheres

Main question: Does every bumpy Riemannian metric on a sphere have at least three short and prime closed geodesics, for some reasonable definition of short? E.g., a geodesic $\gamma$ could be called ...
-1 votes
0 answers
88 views

When is a bundle map between tangent bundles the differential of a function?

I previously asked this question on MSE but did not receive much of a response, so I'll attempt to post it here, edited with the comments received. I hope this question isn't too basic for MO ...
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2 votes
0 answers
76 views

Further directions in representations of surface group into a Lie group

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$I studied the interpretation of Teichmuller space as a representation space for surface groups in $\PSL(2,\mathbb{R})$. Now I am planning to ...
9 votes
2 answers
2k views

Are the models of infinitesimal analysis (philosophically) circular?

Infinitesimal analysis (by which I mean that originating from topos theory---not the nonstandard analysis of Robinson) seeks to recover the pre-limit notions of calculus (which are sufficiently useful ...
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8 votes
1 answer
252 views

Generalized functions in infinite dimensions

What theories are there for generalized functions (distributions) in infinite dimensions? In particular, suppose your "infinite dimensional manifold" is $\mathfrak{M}:=C^\infty(S^1)$. Is ...
2 votes
1 answer
99 views

Vertical Fourier decomposition for skew-Hermitian 1-forms

In an arXiv preprint [2108.05125v1], the authors use the following vertical Fourier decomposition (page 7 therein). Let $(M,g)$ be a Riemannian surface and $SM$ be its unit tangent bundle. Denote by $...
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3 votes
1 answer
147 views

The minimal surface operator in a Riemannian metric

Let $\Omega \subset \mathbf{R}^n$ be a domain, and let the cylinder $\Omega \times \mathbf{R}$ above it be endowed with a Riemannian metric $g$. (Note this is not assumed invariant in the vertical ...
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4 votes
1 answer
95 views

Singular distributions of 1-forms (singular pfaffian systems)

There are several questions in both Math Stack Exchange and Math Overflow about singular distributions of tangent vectors, but I've only found this one about singular distributions of 1-forms (aka ...
4 votes
0 answers
141 views

Physical intuition for curvature on higher order frame bundles?

$\DeclareMathOperator\SO{SO}$A priori: I apologize if this isn't up to Mathoverflow standards, I've had very little luck getting questions on this subject answered elsewhere. I'm looking for a physics ...
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0 answers
48 views

Poincaré-Type inequality for vector fields on the sphere

I have a question which is a vector valued "variant" of the classical Poincaré inequality on the sphere. Consider the sphere $S^{n-1}\subset\mathbb{R^{n}}$ and $\nabla_{s}$ the corresponding ...
4 votes
0 answers
116 views

The homotopy type of the space of symplectic structures

While reading the book Introduction to the $h$-Principle by Y. Eliashberg and N. Mishachev, I noticed that the authors state, at the end of section 9.1.A, that the space of all symplectic structures ...
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0 votes
0 answers
43 views

naturality of the Godbillon-Vey class

This is a problem from Lawrence Conlon's differential manifolds a first course. I do not know how to prove in the following problem If $f: N \rightarrow M$ is transverse to $\mathcal{F}$, prove that $$...
4 votes
2 answers
259 views

Example of a curvature with no associated metric

Is there a concrete example of a $4$ tensor $R_{ijkl}$ with the same symmetries as the Riemannian curvature tensor, i.e. \begin{gather*} R_{ijkl} = - R_{ijlk},\quad R_{ijkl} = R_{jikl},\quad R_{ijkl} =...
2 votes
0 answers
121 views

Minimal Betti numbers of simply-connected threefolds with trivial canonical class

By a threefold, I mean a compact complex manifold of dimension three. For a simply-connected threefold with trivial canonical class, its Betti numbers satisfy: $$b_2 \ge 0, b_3 \ge 2.$$ I am wondering ...
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2 votes
1 answer
122 views

Hyperbolic system of PDEs with elliptic-like boundary contions

Let $\Omega_1$ and $\Omega_2$ be (simply connected) domains on $\mathbb{R}^2$, with coordinates $(x,y)$ and $(X,Y)$ respectively. Given a (smooth) function $Z(X,Y)$ such that $Z\left(\partial \Omega_2 ...
2 votes
0 answers
156 views

Relation between equivariant geometry and representation theory (of geometric objects)

Equivariant geometry studies "manifolds" with an extra structure $G\times M\rightarrow M$. Representation theory studies "Lie algebroids" with an extra structure $\Gamma(M,A)\times ...
1 vote
1 answer
83 views

Curves on $n$-torus analogous to curve implied by diagonal in square for torus

I encountered in my research on dynamical systems a problem, which considers for some $L>0$ on the $C_n=[0,L]^n$ the set $\mathcal{C}_n=\{(x_1,\ldots,x_n)\mid\exists j,k:\,x_{i_j}=x_{i_k}\}$. I am ...
0 votes
1 answer
61 views

Hadamard submanifolds of $k$-fold product of hyperbolic plane

Let $\kappa>0$ and $d,k$ be a positive integers with $k\ge d$. For $k\in \mathbb{Z}^+$ large enough, can one find a geodesically complete and simply connected $d$-dimensional Riemannian ...
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0 votes
0 answers
87 views

Warping a Riemannian manifold until it has non-positive curvature

Let $(M,g)$ be a complete $d$-dimensional Riemannian manifold and let $(\mathbb{H}^2,h)$ be the hyperbolic upper-half plane; and suppose that $\pi_n(M)=\{0\}$ for every $n\in \mathbb{Z}^+$. If $(M,g)$...
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5 votes
1 answer
158 views

Which punctured Riemann surface are the complex structures of complete minimal surfaces in $\mathbb{R}^3$?

Question: Let $\Sigma$ be a punctured Riemann surface(i.e. a closed Riemann surface with several points removed). Is there always a complete conformal minimal immersion $X: \Sigma \to \mathbb{R}^3$? ...
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-5 votes
0 answers
163 views

Why $H^2(X,\mathrm{End}(L))=H^2(X,\mathcal O_X)$?

$\DeclareMathOperator\End{End}$Let $X$ be a compact complex manifold, and $E$ be a holomorphic vector bundle over $X$. In Chan & Suen's paper A differential-geometric approach to deformation of ...
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2 votes
0 answers
111 views

Harmonic functions on varifolds

Let $T$ be a $k$-dimensional varifold in a Riemannian manifold $M$. Assume that $f$ is a smooth function on $M$ which is weakly (sub-)harmonic on $T$; that means that $$ \int \langle \nabla_\omega f, ...
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1 vote
1 answer
157 views

Does the Lie bracket of a certain pair of vector fields vanish?

I'm trying to read section 3 in J. Jost and Y.L. Xin [JX]. This section has to do with the geometry of Grassmannians. I have a question that comes out of the derivation at the top of page 283 in that ...
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2 votes
0 answers
97 views

Strongly constant divergence vector fields

Inspired by this question on homothety vector field we ask the following question Let $M$ be a manifold equiped by a volum form $\Omega$. A strongly constant divergence vector field is a vector ...
1 vote
1 answer
143 views

What is an analogous version of the Ornstein–Uhlenbeck process on Riemannian manifolds?

Recall that the Ornstein–Uhlenbeck (OU) process in $\mathbb{R}^d$ is defined by the following SDE, $$ d Z_t=\frac{-1}{2} Z_t d t+d W_t, \quad t \geqslant 0 $$ where $\left(W_t\right)_{t \geqslant 0}$ ...
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4 votes
1 answer
98 views

Volume of balls in 3-dimensional manifolds with nonpositive Ricci tensor

The volume of a ball in a 3-dimensional Riemannian manifold with nonpositive Ricci tensor is greater or equal to the volume of an Euclidean ball with the same radius? It should not be true, but I am ...
1 vote
2 answers
123 views

Does this special vector field affect on sectional curvature?

We have a nowhere vanishing vector field $X$ on a Reimannian manifold $(M,g)$, such that $\mathcal{L}_X g=2\alpha X^\flat \otimes X^\flat$ for a constant non zero $\alpha$. So, for every vector fields ...
1 vote
0 answers
69 views

Are the level lines of this submersion a trivial fibration?

Let $f$ be the function defined over $$\big\{(x_1,x_2,x_3,x_4)\in\mathbb{R}^4,\,x_1,x_2,x_3>0\big\} \setminus\big\{(x_1,x_2,x_3,x_4)\in\mathbb{R}^4,\,x_1=x_2=x_3+x_4\big\}$$ by \begin{align*} f(\...
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1 vote
0 answers
150 views

Obstruction map for holomorphic line bundle $\operatorname{Ob}_L:H^1(X,\mathcal O)\to H^2(X,\mathcal O)$

$\DeclareMathOperator\Ob{Ob}$In Chan & Suen's paper A differential-geometric approach to deformation of pairs $(X, E)$ p.20, the authors define a Kuranishi map (or obstruction map) for the ...
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