Questions tagged [dg.differential-geometry]

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

Filter by
Sorted by
Tagged with
0 votes
0 answers
5 views

Handling Degenerate Planes in Pseudo-Riemannian Geometry: Impact on Sectional Curvature and Comparison Theorems

I've been studying Riemannian and pseudo-Riemannian manifolds and came across an intriguing point regarding the definition of sectional curvature in both geometries. In pseudo-Riemannian geometry, for ...
lming2's user avatar
  • 43
2 votes
1 answer
109 views

Relationship between Frobenius theorem, curvature, and integrability

In this answer to References for "modern" proof of Newlander-Nirenberg Theorem John Hubbard alluded to something called the "Frobenius integrability form" $\phi\mapsto\bar\partial\...
level1807's user avatar
  • 467
1 vote
0 answers
68 views

Extend a circle action on $3$-manifolds

Let $M$ be an oriented closed $3$-manifold equipped with an effective smooth circle action. Can we have a classification of all such $M$ such that there exists a $4$-manifold $N$ with $\partial N=M$, ...
Zhiqiang's user avatar
  • 881
4 votes
0 answers
83 views

Regularity of exponential map for $C^{2,\alpha}$ Riemannian metrics

Let $g$ be a $C^{2,\alpha}$ Riemannian metric and $0<\alpha<1$. Would the exponential map $\mathrm{exp}_p$ be $C^{1,\alpha}$ as the point $p$ varies? Since $\mathrm{exp}_p$ is defined by the ...
user486255's user avatar
0 votes
0 answers
32 views

Construct compact submanifold containing non-compact Nash embedded submanifold

$$ \newcommand{\R}{\mathbb{R}} \newcommand{\geu}{g_{\text{Eu}}} \newcommand{\X}{\mathcal{X}} \newcommand{\iX}{\mathring{\X}}$$ Let $\X$ be a closed bounded convex set in some Euclidean space. Its ...
DavideL's user avatar
  • 111
13 votes
0 answers
217 views

Which G-structures lead to analyticity?

It’s well known that integrability conditions on $G$-structures sometimes force a unique analytic structure on the underlying manifold: A Lie group structure (a 1-structure with equivariantly ...
level1807's user avatar
  • 467
2 votes
0 answers
38 views

Under what conditions principal directions define an integrable distribution?

Consider a hypersurface $M^n \subset \mathbb{R}^{n+1}$ which is compact without boundary. Assume that its second fundamental form $A$ has distinct eigenvalues $\lambda_1<\ldots<\lambda_k$ (with $...
Dorian's user avatar
  • 331
5 votes
0 answers
48 views

The universal cover and embedding

Whether there exists a non-simply connected complete Riemannian manifold $(M^n, g)$ such that it can be $C^2$-isometric (globally) embedded into some Euclidean space $\mathbb{E}^p$, but its ...
Jialong Deng's user avatar
  • 1,749
1 vote
0 answers
64 views

Metric of negative curvature on connected sum

Let $(M_1,g_1)$ and $(M_2,g_2)$ be two Riemannian manifolds of dimension $n\geq 2$. If we consider the connected sum $M=M_1\mathbin{\#}M_2$ of the two manifolds; can one get a smooth metric $g$ on $M$ ...
User5's user avatar
  • 11
3 votes
1 answer
179 views

A complex version of the Cahiers topos

Has anyone tried defining a complex version of the Cahiers topos? If we take the definition of $C^\infty$-rings, replace "smooth" with "holomorphic" (of course, one has to take ...
xuq01's user avatar
  • 1,032
2 votes
0 answers
189 views
+50

Do the eigenvalues of a thin strip around a semicircle converge to the respective eigenvalues of a semicircle?

Background: Consider the sphere $S^2$ with the round metric $g$ and let $\gamma$ be one half of a great circle of length $\pi.$ Let $T_\epsilon$ denote a geodesically convex tube around $\gamma$ of ...
Student's user avatar
  • 601
2 votes
0 answers
33 views

Growth/Decay of conformal Killing fields in cone metrics

Let $\gamma$ be a smooth metric on $S^2$ of positive curvature. Consider the metric $$g= dr^2 + r^2 \gamma$$ on $[1,\infty) \times S^2$. Does there exist a nontrivial conformal Killing field vanishing ...
Laithy's user avatar
  • 865
2 votes
1 answer
136 views

Gluing local holomorphic connections

On page $180$ of Complex Geometry by Daniel Huybrecths, he defines the so called Atiyah class of a holomorphic vector bundle by the Čech cocycle $$A(E)=\{U_{ij}, \psi^{-1}_j \circ (\psi^{-1}_{ij}d\...
Johannes's user avatar
2 votes
1 answer
85 views

For a closed Riemannian manifold $M$, must the set of points with non-unique closest points to a closed submanifold $S$ of $M$ be of 0 volume measure?

Let $(M,g)$ be a closed (compact without boundary) Riemannian manifold of finite dimension, with the volume measure $\mu:= \mu(E):=\int_{E}dvol_g \forall E \in \mathcal{B}(M),$ the Borel sigma algebra ...
Learning math's user avatar
-1 votes
0 answers
77 views

Zero of gradient of a sum of squares of polynomial

Let $f_1,\ldots,f_n\in \mathbb{R}[x_1,\ldots,x_n]$ have non zero constant jacobian determinant. Then is it true that $f_1^2+\cdots+f_n^2$ has any critical point? Let me explain a motivation. If ...
George's user avatar
  • 449
0 votes
0 answers
138 views

Kähler manifold with negative sectional curvature

Goldberg's theorem states that every almost Kähler manifold of constant curvature is Kähler if and only if the curvature is zero. This seems to contradict the fact that the sectional curvature of the ...
Samir's user avatar
  • 43
1 vote
0 answers
68 views

Total curvature of a conjugate minimal surface

Let $s: S \to \Bbb R^3$ be an immersed minimal surface with finite total curvature and a proper annular end (possibly with other types of ends). What is exactly meant by a proper annular end? It is an ...
Annetta's user avatar
  • 11
3 votes
1 answer
240 views

Ricci flow and curvature

I am trying to read about geometric flows mainly Ricci flows. I have a question in mind, which I am not sure whether it's possible or not. So my question is if one starts with a metric that has mostly ...
Emmie's user avatar
  • 31
2 votes
1 answer
166 views

Frobenius theorem and the size of integral manifold

Let $X =(X_0,X_1)\in \mathbb{R}^2$ and $Y=(Y_0,Y_1)\in \mathbb{R}^2$ be two vector fields on $\mathbb{R}^2$ such that $X,Y$ are independent on each tangent plane and $[X,Y]:=XY-YX=0$. Then by ...
George's user avatar
  • 449
2 votes
1 answer
81 views

Parameterizing Teichmüller spaces of punctured surfaces

Let $S_{g,n}$ denote a genus $g$ surface with $n$ punctures. There is a map $F$ from the Teichmüller space of the punctured surface $T(S_{g,n})$ to the Teichmüller space of the compact surface $T(S_{g}...
Yousuf Soliman's user avatar
9 votes
0 answers
303 views

Mappings of the sphere (to itself) defined by homogeneous polynomials

Preamble $\DeclareMathOperator\SO{SO}$Let $\mathbb{S}^m\subset \mathbb{R}^{m+1}$ be the standard unit sphere. An observation of Do Carmo and Wallach states that If $G$ is a subgroup of $\SO(m+1)$ ...
Willie Wong's user avatar
  • 36.9k
1 vote
0 answers
81 views

Describing the hyperbolic structure of punctured torus in terms of the period lattice

Let $T$ be a torus, $T^* = T - \{p\}$ be the complement of a point $p$. Let's fix a pair of generators $x,y\in\pi_1(T^*)$. Their images in $\pi_1(T)$ also generate, and will also be denoted by $x,y$. ...
stupid_question_bot's user avatar
1 vote
0 answers
62 views

Extending $G$-equivariant local diffeomorphisms on principal bundles to local bundle maps

Consider a principal $G$-bundle $P$ over the base space $M$ equipped with a connection 1-form $\omega$. Let $\mathcal{U}$ and $\mathcal{V}$ be open subsets of $P$, and suppose $F: \mathcal{U} \to \...
Amin's user avatar
  • 83
5 votes
1 answer
221 views

Bochner Laplacian in coordinates

Sorry if this is a too basic question, but I didn't find an answer anywhere: The connection Laplacian, or Bochner Laplacian, is the differential operator acting on $k$-tensor fields $T\in\Gamma^{\...
B.Hueber's user avatar
  • 987
0 votes
0 answers
42 views

How do I find an algebraic expression for the function $F(ξ, \bar{ξ})$ from this paper?

I am working on understanding the paper "On $C^2$-smooth Surfaces of Constant Width" by Brendan Guilfoyle and Wilhelm Klingenberg. As part of their definition of equations for a 3D surface, ...
Lawton's user avatar
  • 105
2 votes
0 answers
45 views

Smoothness of the Fréchet Function

Let $M$ be a compact Riemannian manifold and $d$ be the induced distance function. Suppose $\mu$ is a probability measure on $M$ with continuous density. The Fréchet function is defined as $$ F(x) = \...
Yueqi's user avatar
  • 21
7 votes
3 answers
598 views

A continuous version of Carathéodory's convex hull theorem

A well-known theorem of Caratheodory states that any point in the convex hull of a set $X\subset R^n$ lies in the convex hull of at most $n+1$ points of $X$. I am wondering about a version of this ...
Mohammad Ghomi's user avatar
3 votes
0 answers
91 views

On the linearized evolution equations in general relativity

The following puzzles me already for quite some time: In mathematical relativity, especially in the discussion of the Cauchy problem, one usually works in the so-called ADM-Formalism, in which one ...
G. Blaickner's user avatar
  • 1,137
5 votes
1 answer
121 views

Regularity requirements for Sard's Theorem

The most common formulation of Sard's Theorem is that for $f\in C^{n-m+1}(\mathbb R^n, \mathbb R^m)$ with $n\ge m$, the set $f(C_f)$ has Lebesgue measure 0, where $C_f=\{x, df(x)=0\}$. Question. Is it ...
Bazin's user avatar
  • 15k
0 votes
0 answers
20 views

Is it possible to find the intersection of this involute and roulette, given their parametric equations?

Background I have two parametric curves, and I want to find the parameter values of their intersection point closest to zero under certain conditions. The first curve is an involute of a circle with ...
Lawton's user avatar
  • 105
4 votes
0 answers
287 views

Milnor’s smoothed corners technique for a product of manifolds with boundary and boundary defining functions

Let $M$ be a smooth manifold that we view as the interior of a compact manifold with boundary $\overline{M}$. Let $\rho$ be a boundary defining function for $\overline{M}$, i.e. $\rho$ is smooth, $\...
zarathustra's user avatar
4 votes
0 answers
383 views

A 4th-order linear PDE

I am interested in the following type of $4$-th order linear PDE with $2$ variables (i.e., $x$ and $t$): $x^3 f_{xxxt}+ f =0$ Does anyone know if this type of PDE already appeared in the literature? ...
Math2024's user avatar
1 vote
0 answers
60 views

Does the local family index theorem hold for compact manifolds with corners?

Let $\pi:X\to B$ be a submersion with closed, oriented and spin fibers of even dimension. Suppose $X$ and $B$ are compact, and let $E\to X$ be a complex vector bundle over with a Hermitian metric $g^E$...
Ho Man-Ho's user avatar
  • 1,087
2 votes
0 answers
89 views
+50

soft question, proof sketch: Constructing spaces of analytic sections. silva spaces

This is a follow up on this question Representing geodesic compactifications of $S^1\times \Bbb R$ as analytic sections over base (analytic) foliations to add more details. In the linked question I ...
53Demonslayer's user avatar
2 votes
1 answer
83 views

Coupling small and large injectivity radii

I'd like to know whether a manifold of constant curvature, which has large injectivity radius at many points, can have points of arbitrary small injectivity radius. More precisely, for a point $x$ in ...
Nandor's user avatar
  • 289
5 votes
0 answers
130 views

Riemannian structure on connected Hilbert manifolds

The infinite-dimensional separable Hilbert space $H$ has the unusual property that it is diffeomorphic to its unit sphere $S^{\infty}$. Therefore, $H$ admits the round metric as a complete and bounded ...
Zerox's user avatar
  • 1,111
1 vote
0 answers
66 views

Representing geodesic compactifications of $S^1\times \Bbb R$ as analytic sections over base (analytic) foliations

Given a smooth nested set of "partial" foliations $\mathcal F_{\alpha}=\big\lbrace e^{\frac{\alpha}{\log x}}: \alpha \in (1/k,k), x\in(0,1),k\in [1,\infty) \big\rbrace$ of $X^2=(0,1)^2$ with ...
53Demonslayer's user avatar
0 votes
0 answers
72 views

Hyperbolicity in sense of Koszul

Question : I have a question about the notion of hyperbolicity in the sense of Koszul. Koszul asserts that locally flat compact hyperbolic manifolds do not contain a parallel vector field. Do you have ...
Emmanuel Gnandi's user avatar
4 votes
1 answer
133 views

Can every surface be realized as a mean convex hypersurface in $\mathbb{R}^3$?

I'm wondering if every closed surface can be realized as a mean convex hypersurface in $\mathbb{R}^3$, i.e. the mean curvature vanishes or points inward. Categorizing by genus: for $S^2$ ($g = 0$) ...
JMK's user avatar
  • 301
1 vote
0 answers
44 views

Elliptic surfaces with monodromy in Borel subgroup

Are there restrictions on the invariants of an elliptic surface $M\overset{\pi}{\longrightarrow} C$ for the monodromy of its homological invariant to be contained in the upper triangular subgroup of $\...
AG14's user avatar
  • 59
2 votes
0 answers
51 views

Is a $C^1$ surface with $C^1$ boundary and uniformly continuous normal a manifold with boundary?

Consider an oriented $C^1$ surface ${\cal S}$ whose closure is a $C^0$ surface with boundary whose boundary is a $C^1$ curve. If the normal to ${\cal S}$ is uniformly continuous, so that it has a ...
Brian Seguin's user avatar
2 votes
0 answers
157 views

When is the closure of a manifold a manifold with boundary?

I'm looking for conditions that ensure the closure of an embedded manifold is a manifold with boundary. The specific case I'm interested is as follows. Consider an oriented $C^1$ surface ${\cal S}$ in ...
Brian Seguin's user avatar
1 vote
0 answers
45 views

Using connection form for unknown frame field

I have a way to calculate the connection 1-form $\alpha$ associated to a compact simply connected parallelizable Riemannian surface $(M,g)$ (so, $M$ is topologically a disk) and a special orthonormal ...
yak's user avatar
  • 11
-4 votes
1 answer
301 views

Does a coarser topology lead to a non-Hausdorff topological manifold? [closed]

Take a topological manifold $M$. Suppose one considers a strictly coarser topology than the manifold topology. Can such topology result in a non-Hausdorff topological manifold? NOTE: PLEASE avoid the ...
Bastam Tajik's user avatar
-2 votes
1 answer
112 views

Is this single-variable function being called with two variables (and if so, how do I handle that), or am I misreading the notation? [closed]

While working on a project, I came across a paper that includes this sum on page 15 as the definition of a support function $r$ for a surface with tetrahedral symmetry: $$ r(ξ, \bar{ξ}) = \frac{1}{\#𝒢...
Lawton's user avatar
  • 105
7 votes
1 answer
489 views

Conformal Killing fields satisfy a third order PDE

Let $(M,g)$ be a $n$-dimensional Riemannian manifold with smooth metric $g$. Proposition 3.2 in the paper "The Boost Problem in General Relativity" by O'Murchadha and Chistodoulou claims ...
Laithy's user avatar
  • 865
13 votes
3 answers
2k views

How to get to the earliest time zone?

You are in a plane at some point on Earth. You want to be at the earliest time zone possible at the end of your flight. What is the optimal path to take? Formally, fix spherical coordinates $(\theta, \...
Nate River's user avatar
  • 4,802
2 votes
1 answer
226 views

How to chart tubes around manifolds with boundary/corners?

Let $M \subset \mathbb{R}^d$ be a manifold with boundary/corners. For example, a piece of curve with endpoints or a $2d$ unit square in $\{ z = 0 \}$. I am interested in introducing local coordinates ...
tsnao's user avatar
  • 462
2 votes
1 answer
182 views

Does the group of compactly supported diffeomorphisms have the homotopy type of a CW complex?

It is known that the group of diffeomorphisms of a compact manifold with the natural $C^{\infty}$ topology has the homotopy type of a countable CW complex. See for instance this thread: Is the space ...
Yasha's user avatar
  • 469
2 votes
0 answers
372 views

Parallel transport on a vector bundle : expansion of the correspondance between normal and tubular coordinates

Let $(M, g^{TM})$ be a Riemannian manifold of dimension $n$. Let $X \subset M$ be a submanifold of dimension $n$ with boundary $\partial X$. Then we have a splitting of the tangent bundle $$TM \vert_{\...
hseldon39's user avatar

1
2 3 4 5
173