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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Naturality of Lie bracket - alternate proof
Let $M$ and $N$ be smooth manifolds, and let $F: M \to N$ be a smooth map. Let $X$ and $Y$ be vector fields on $M$, and let $\tilde{X}$ and $\tilde{Y}$ be vector fields on $N$. We say that $X$ and $\...
-3
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0
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96
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On unitary manifold [closed]
For integer $d$, we can define unitary matrices $\mathbb{U}:=\{U\mid U^*U=UU^*=I_d\}$. This forms a manifold of dimension $d^2$.
Suppose we are given a submanifold of $\mathbb{U}$, denoted as $\mathbb{...
0
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0
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73
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Can a Tangent Space always be expressed with “more structure” than just a vector space (e.g. a choice of basis for Stiefel manifold)
I'm currently trying to read about the Stiefel manifold, or set of all $p$ orthonormal $n$-dimensional vectors embedded in $\mathbb{R}^{n\times p}$.
$$\mathcal{V}_p(\mathbb{R}^n) = \{U \in \mathbb{R}^{...
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0
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83
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Expressing the union of principal orbits as a disjoint union of global slices for proper group actions
Setup:
I was reading about slices and principal orbit theorems (Theorem 3.4.6) from these notes.
Let the Lie group $G$ act on a complete Riemannian manifold $(M,g)$ isometrically on $M$, i.e. $\phi^{*}...
5
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0
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46
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Approximation of triply periodic minimal surfaces with trigonometric level sets
Some triply periodic minimal surfaces are known to be approximated by trigonometric level sets very accurately.
To see this, let's sample a gyroid scaled to the bounding box $[0, 1]^3$ exactly through ...
1
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0
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78
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Compactification of a Cartan-Hadamard manifold
Let $X$ be a simply connected manifold with nonpositive sectional curvature. It is standard that $X$ is uniquely geodesic, i.e., for any distinct points $p$ and $q$, there is a unique geodesic ...
0
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37
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Prove the orthogonality of vector spherical harmonics
We define
$S_a^{lm} = \Big( - \frac{1}{\sin \theta} Y^{lm}_{,\varphi}, \sin \theta\ Y^{lm}_{,\theta} \Big)$
$Y_a^{lm} = \Big( Y^{lm}_{,\theta}, Y^{lm}_{,\varphi} \Big)$
to be the axial vector ...
6
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1
answer
150
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Coarse embeddings and Gromov products in (Gromov) hyperbolic spaces
I am new into geometric group theory and I have recently started reading the book "Sur les Groupes Hyperboliques d’après Mikhael Gromov" by Ghys and de la Harpe. The following inequality ...
5
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0
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197
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Theories of manifolds w/ extra structure and singularities
Many different objects in mathematics can be described as manifolds with extra structure. Among the most famous examples of these are smooth manifolds, Riemannian manifolds, complex manifolds, and ...
1
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1
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141
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Proving that $H^1(X,\mathcal{Hom}(\mathcal{G},\mathcal{E})) \cong \text{Ext}^1(\mathcal{G},\mathcal{E})$ holds for locally free sheaves
The following passage is from a thesis I'm reading:
Suppose we have a short exact sequence of vector bundles $$0 \to \mathcal{E} \to \mathcal{F}\to \mathcal{G} \to 0.$$ Since these sheaves are ...
3
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1
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+100
Systems of (hyperbolic) 2nd order PDEs with lower order constraints
Certain surfaces in mechanics are endowed with the fundamental forms
\begin{align}
\text{I} &= \mathrm{d}u^2+\mathrm{d}v^2+2\cos\gamma\: \mathrm{d}u\: \mathrm{d}v \\
\text{II} &= \alpha\left(\...
0
votes
1
answer
99
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Regular value theorem for Banach manifolds without surjectivity
It is well-known (e.g. Lang, Fundamentals of differential geometry, Prop. 2.3 in Chapter II) that the following extension of the regular value theorem holds for Banach manifolds:
Let $\phi : M\...
0
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1
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118
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Vector bundles over a homotopy-equivalent fibration
I think this question is related to what is known as "obstruction theory", but I'm not very familiar with this field of mathematics, so I am asking here.
Let $\pi:N\rightarrow M$ be a smooth ...
0
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0
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65
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Convergence of metric implies convergence of eigenvalues?
Consider the metric $g_\varepsilon=d\theta^2 + \varepsilon^2 \sin(\theta)^2 \, d\varphi^2$ on the hemisphere $S^2_+.$ I had two naive questions:
Does $g_\varepsilon$ converge to the flat metric on ...
6
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1
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Densities, pseudoforms, absolute differential forms and measures, differential forms, etc
Apologies if this question is too basic, but I figured I first heard of most of these concepts on MO, so perhaps I can ask here.
Gelfand’s definition, copied from AlvarezPaiva [My edit, could be ...
1
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0
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97
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Dolbeault class of the curvature of the Chern connection equaling the Atiyah class
The following is a proposition from Complex Geometry by Huybrechts.
Proposition $4.3.10$. For the curvature $F_\nabla$ of the Chern connection on an hermitian holomorphic vector bundle $(E,h)$ one ...
3
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0
answers
61
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Surface terms in the calculus of variations on jet bundles
Let $\pi:N\rightarrow M$ be a fibered manifold with $m=\dim M$ and $m+n=\dim N$. The variational bicomplex on the infinite jet space $J^\infty(\pi)$ is denoted $(\Omega^{k,l}(\pi),\delta,\mathbf d)$ ...
1
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1
answer
107
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Does a Riemannian submersion map horizontal geodesics to geodesics, and a relevant question?
I asked this question on MSE, but I didn't receive a response yet, so I'm asking here. Apologies if the question is not exactly a research level question, but I'm having some trouble in figuring them ...
0
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1
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45
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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 ...
2
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2
answers
165
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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\...
1
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0
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84
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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$, ...
4
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0
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89
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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 ...
0
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0
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36
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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 ...
13
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0
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254
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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 ...
2
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0
answers
42
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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 $...
5
votes
0
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49
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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 ...
1
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0
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71
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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$ ...
4
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1
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188
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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 ...
2
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0
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233
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Convergence of metric and eigenvalues on a tubular neighbourhood
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 ...
2
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0
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45
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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$$
on $[1,\infty) \times S^2$.
Does there exist a nontrivial conformal Killing field vanishing ...
2
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1
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150
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Gluing local holomorphic connections
On page $180$ of Complex Geometry by Daniel Huybrecths, he defines the so called Atiyah class of a holomorphic vector bundle by the Čech cocycle $$A(E)=\{U_{ij}, \psi^{-1}_j \circ (\psi^{-1}_{ij}d\...
2
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1
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89
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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 ...
0
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0
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139
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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 ...
1
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0
answers
72
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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 ...
3
votes
1
answer
251
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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 ...
2
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1
answer
178
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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 ...
2
votes
1
answer
84
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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}...
9
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0
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317
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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 $\SO(m+1)$ ...
1
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0
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85
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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 $x,y$.
...
1
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0
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62
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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 \...
5
votes
1
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230
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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^{\...
0
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0
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44
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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, ...
2
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0
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46
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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) = \...
7
votes
3
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605
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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 ...
3
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0
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92
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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 ...
5
votes
1
answer
123
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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 ...
0
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0
answers
21
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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 ...
4
votes
0
answers
289
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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, $\...
4
votes
0
answers
412
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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? ...
1
vote
0
answers
60
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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 $g^E$...