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Questions tagged [global-analysis]

In mathematics, global analysis, also called analysis on manifolds, is the study of the global and topological properties of differential equations on manifolds and vector space bundles.

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examples of non-flat, asymptotically flat manifold with non-positive curvature

I would like to know if there are any natural (e.g., physical) examples of non-flat, asymptotically flat manifold with non-positive sectional curvature? For example, any minimal surface of such a ...
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Rank of smooth distribution

Let D is a smooth distribution of vector bundle $\pi: E\rightarrow M$ such that M is a manifold. Why rank of D is lower semi-continuous function on a M?
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Computation involving codifferential and Hodge star

Let $(M,g)$ be an oriented Riemannian manifold. Then, the codifferential $\delta$ is given by $\delta \omega=-\star d \star \omega$, where $\star$ stands for the Hodge star, $d$ for the exterior ...
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25 views

geodesic balls in Riemannian manifolds with bounded geometry

Let $(M,g)$ be an open (:=complete, non-compact) Riemannian manifold with bounded geometry, in the sense that in some atlas of charts of radius $r_0>0$, the metric and all its derivatives are ...
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Is the functional $w \mapsto \int_0^1 | \ \| w(t) \| - 1 | \ dt$ $C^1$ or even smooth?

Let $H:= H(I;\mathbb{R}^3)$ be the space of $L^2$ + absolutely continuous functions with $L^2$ derivative. For $w \in H$ consider the functional $$\psi(w) = \int_0^1 | \ \| w(t) \|^2 - 1 | \ dt$$ ...
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Ordinary and covariant derivative inequality: $ \| u\|_1 \leq C\left( \| \frac{Du}{dt} \|_0 + \|u\|_0 \right) $

Let $I=[0,1]$. Define: $H_0 = L^2(I,\mathbb{R}^3)$ with inner product $\langle u,v \rangle_0 = \int_0^1 \langle u(t), v(t) \rangle \text{ dt}$ $H_1 = W^{1,2}(I, \mathbb{R}^3)$ with inner product $\...
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40 views

Curvature of a homogenous manifold.

I was a reading a paper and it seemed to me that in one of the equations the authors used the fact if $M$ is a homogenous Riemannian manifold (i.e., the group of isometries of $M$ act transitively on $...
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51 views

Orientation forms on compact smooth manifolds are equivalenf if they have the same integral

I have a problem solving the following exercise (Ex. 22-15) from John M. Lee‘s „Introduction to smooth manifolds“. The problem is the following: Use the same technique as in the proof of the Darboux ...
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134 views

Confusion about definition of continuous spectrum

In discussions about the spectrum of hyperbolic surfaces, people seem to be interested in 'eigenvalues embedded in the continuous spectrum'. I am wondering which definition of the term 'continuous ...
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100 views

Global Implicit Function Theorem

I encounter the problem that I would like to extend the implicit function theorem (for real numbers) to a global version. The classical implicit function theorem is given by the following: Assume $F: ...
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Generalisation of the Hopf–Rinow theorem

Is their a generalisation of the Hopf–Rinow theorem to infinite dimensional manifolds (Frechet, Banach or Hilbert)? I have actually never seen the prove of the Hopf–Rinow theorem so i don't know if ...
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108 views

Vector-valued forms inside the first jet bundle

On page 433 of "Self-duality in four-dimensional Riemannian geometry" by Atiyah, Hitchin and Singer, it is written that $p^*(E \otimes \Lambda^1) \subset p^*J_1(E)$, where $\Lambda^1 \to X$ is the ...
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What are examples of a second order operational tangent vector on an infinite dimensional Hilbert space.

In the book "a convenient setting for global analysis" they describe the order of an operational tangent vector on a convenient vector space. http://www.mat.univie.ac.at/~michor/apbookh-ams.pdf ...
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60 views

non-negative curvature of space curve

I read in most of the textbook that "curvature of a space curve is always non-negative" but I could not understand the intuition behind this that why is so? Give some nice intuition and proof.
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50 views

Existence of canonical connection on trivial Banach bundle

Let $W$ be a (real) Banach space and $M$ be a (finite-dimensional) Manifold. Consider the trivial $W$-bundle $\pi: E \to M$ over $M$ with $E := W \times M$ and $\pi$ the second-factor projection. ...
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43 views

Diffeomorphisms and their derivatives

If a $C^k$ map $f: X\to Y$ where $X, Y$ are Banach spaces is a diffeomorphism. What can we say about $d_xf$, the differential of $f$ at $x\in X$? It is true that we have $$ \inf_{x\in X}\big|\det [...
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Has this functional been studied somewhere?

$\newcommand{\M}{\mathcal{M}}$ $\newcommand{\N}{\mathcal{N}}$ $\newcommand{\TM}{\operatorname{T\M}}$ $\newcommand{\TN}{\operatorname{T\N}}$ Let $\M,\N$ be Riemannian manifolds, $f:\M \to \N$ smooth. ...
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1answer
78 views

Why harmonicity is a local property?

Given two Riemannian manifolds $M,N$, we say that $f:M \to N$ is harmonic if it is a critical point of the Dirichlet energy functional. More precisely, this means that for every variation $f_t$ of $...
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1answer
309 views

one parameter family of diffeomorphism $\phi_{t}$ of $\mathbb{R}^2$

I have the following question: The one parameter family of diffeomorphism $\phi_{t}$ of $\mathbb{R}^2$ to itself for $t\in (\pi,\pi)$ is defined in polar coordinates $(r,\theta)$ by $$\phi_t(r,\...
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1answer
27 views

Let $G$ act smoothly on $M$, $(g,m) \rightarrow t_g (m) $ from the left. Why is $t_g \simeq id $ if $G$ connected?

Let $G$ act smoothly on $M$, $(g,m) \rightarrow t_g (m) $ from the left. Why is $t_g \simeq id $ if $G$ connected? I've seen this statement, but have no idea how to use connectedness here. Does anyone ...
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$G$ is a Lie group and $M$ a manifold. What does it mean that $G$ acts as a group of automorphism on $H^p (M, \mathbb{R} )$?

I stumbled upon this in Bredon and don't know how to understand this. What I have given is a smooth action from $G$ on $M$. I did not find any special definiton for the above phrase, so I expect an ...
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Let a compact group $G$ act on a manifold $M$. Why does G act as a group of automorphisms on $H^p (M, \mathbb{R} )?

Let $(\sigma , m) \rightarrow t_{\sigma}$ be this action. A definition for acting on a group of automorphisms I found here: Action via automorphism. t^{*}_{\sigma} seems to be this action and I can ...
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Can one always find a basis of global vector fields $X_0 , \ldots , X_n$ for the tangent space $T_p M$ for a manifold $M$? [closed]

I know that locally that is possible, as for local coordinates $x_i$ $(\frac{\partial}{\partial x_i})_{i \in I}$ spans $T_p M$ and therefore, one gets a local vector field. How do I extend this one?
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Showing that $\bar\partial_J$ is a smooth section of Banach bundle

I am reading Chapter 3 (Moduli Spaces and Transversality) of "J-holomorphic curves and symplectic topology" by McDuff & Salamon. Fixing $k\in\mathbb N,p>1$ such that $kp>2$, they consider $\...
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294 views

Compute the first distributional derivative of a function

Consider $\Omega = (0,2) \subseteq \mathbb{R} $ to be a support of the function $f(x)$ defined as $$ f(x) = \begin{cases} x, &0 < x \leq 1\\ 2, &1< x \leq 2 \end{cases}$$ I need to ...
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1answer
75 views

Show that $\int_{\mathbb{R}} \phi (x) dx= 0$

Let $\phi \in C_{c}^{\infty}\mathbb{(R)}$ . I need to show that $\int_{\mathbb{R}} \phi (x) dx = 0$ iff there exists a function $\psi \in C_{c}^{\infty}\mathbb{(R)}$ such that $\phi(x) = \psi ' (x)$. ...
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183 views

Friedrich extension of an elliptic operator

I'm working with elliptic operators on Riemannian manifolds. In the case where the metric $g$ on $(M,g)$ is not complete, I understand that, for example, the Hodge laplacian may be not essentially ...
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A question regarding mollifers on Sobolev spaces on closed manifolds

Let $M$ be a closed Riemannian manifold and denote by $H^s(M), \, s\in \mathbb{R} $ the standard Sobolev spaces on $M$ defined using powers of $1+\triangle$. Let $J_n: \mathcal{D}'(M)\rightarrow \...
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Difference between Homogeneous first order operator and Homogeneous first order differential operator

By definition, a Homogeneous first order operator on the algebra of differentiable functions is an operator like $$ Df = \sum \phi_i \frac{\partial f}{\partial x_i} $$ where $\phi_i$ are ...
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1answer
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Stuck at showing that $g_*f_* = (gf)_*$

Let $X,Y,Z$ be topological spaces and $f : X \rightarrow Y , g: Y \rightarrow Z$ be continuous functions. Consider $Sh(X) ,Sh(Y),Sh(Z)$ to be the set of sheaves over $X,Y,Z$ respectively. Now we ...
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How to compute one-parameter group and corresponding vector fields

I have two related questions to ask - $1)$ Let $\rho : \mathbb{R} \rightarrow G$ be a one-parameter group. ($\mathbb{R}$ and $G$ are Lie groups). If we take $G = S^1$ then the left invariant vector ...
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1answer
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Show a sequence of sheaves is exact

Given $f : X \rightarrow Y$ , a continuous map, let $0 \rightarrow \Im \rightarrow \mathcal{F} \rightarrow \mathcal{H} \rightarrow 0$ be an exact sequence of sheaves on $Y$. I need to prove that $$ 0 ...
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1answer
35 views

Show bijective correspondence

Let $f : X \rightarrow Y$ be a continuous map. Let $\Im$ be a sheaf on $Y$ and $U \subseteq Y$ be an open subset. I need to show that there is a bijective correspondence between $(f^{-1} \Im)(f^{-1}(U)...
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37 views

Clarification in the definition of presheaf

I need a clarification in the definition of presheaf. If $X$ is a topological space then we define presheaf on $X$ to be an assignment to every open subset $U \subset X$ and to every pair of open sets ...
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2answers
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Prove that a set of differentiable function on $\mathbb{R^n}$ is a sheaf

Let $X = \mathbb{R^n}$. For every open subset $U \subseteq X$ , consider a presheaf $ , \Gamma(U) := \{ f: U \rightarrow \mathbb{R} : f $ is differentiable$ \}$. I need to show that this presheaf is ...
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1answer
36 views

Restriction of a null set

Suppose you have a null set, $S$ in $\mathbb{R}^n$. Is it true than in that case, there always exists an immersion $i: \mathbb{R} \hookrightarrow \mathbb{R}^n $ such that for almost all $x \in \mathbb{...
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1answer
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How to show the space of closed curve is Hilbert manifold?

In the picture below ,$(M,g)$ is a Riemannian manifold. Why $\mathcal L_M$ is a Hilbert submanifold of $L^{1,2}(S^1,R^r)$ ? Besides, what is the inner and name of $L^{1,2}(S^1,R^r)$ ? The picture ...
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What are the differences among different notions of harmonic maps?

Given smooth (compact, if needed) Riemannian manifolds $M$ and $N$. There are at least 3 different notions of harmonic maps (shortly, HM): weakly HM. stationary HM. minimizing HM. It is well-known ...
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Condition for Dirichlet boundary conditions

Let D be a differential operator on a manifold with boundary. We consider the differential equation $Df = 0$ with Dirichlet-boundary-conditions $f|_{\partial M}= g$. Are there cases, where not every ...
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How far can we push the Schwartz kernel theorem?

The Schwartz kernel theorem works for operators defined on $C_ {c}(\mathbb{R}^n,E)$, as long as $E$ is finite-dimensional and we introduce the right notion of a generalised section. In every ...
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1answer
562 views

Nonexistence of conjugate points $\Rightarrow$ a geodesic is minimizing

Motivation: I am trying to prove a certian geodesic is minimizing. The only generic tool I know for doing that is the fact the a gedoesic is minimizing as long as it stays in a normal neighbourhood of ...
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1answer
135 views

An stronger form of the existence of a smooth Urysohn function on $\mathbb R^n$

I proved the following form of the existence of a smooth Urysohn function:: proposition: For any compact set $K\subset\mathbb R^n$ and any open set $U\subset\mathbb R^n$ where $K\subset U$, there is ...
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1answer
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The existence of a smooth functions taking values $0$ and $1$ on two given closed sets

In theorem 5.1 on page 39 Boothby's book(An introduction to Differentiable Manifolds By William M. Boothby), he prove that: Let $F\subset \mathbb R^n$ be a closed set and $K\subset \mathbb R^n$ ...
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asking a way to prove an inequality

Assume $\Omega$ is a bounded smooth domain in $\mathbb R^N $ with $N \ge 5 $ and $u \in C^2(\Omega)$ . I want to proof $$\int_{\Omega}\frac{|\nabla u|^2}{|x|^2}d{x} \;\ge\; \left(\frac{N-4}{2}\...
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1answer
263 views

Reference request: infinite-dimensional manifolds

The following books and/or notes develop various aspects of the theory of infinite-dimensional manifolds: Lang, Fundamentals of Differential Geometry. Kriegl & Michor, The Convenient Setting of ...
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1answer
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When are heat kernels only dependent on the distance?

"All" the examples of heat kernels in circulation are only dependent on the distance between the space variables rather than on the space variables themselves, i.e. $$K(t;x,y) = K(t;d(x,y)).$$ Think ...
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What's wrong in this prop about volume form if we drop “oriented”?

I was studying Prop 15.29 from Lee's Introduction to Smooth Manifold and I asked myself what's wrong with this proof if we drop the oriented assumption. I know that I'd came up with a non zero $n$-...
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1answer
969 views

Hodge star operator and volume form, basic properties

let $(M,g)$ be an oriented Riemannian manifold. Let $*$ be the hodge operator, I want to prove that $$*\mathrm{vol}_g =1$$ where $\mathrm{vol}_g$ is the associate volume form $\sqrt{g} e^1\wedge \...
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1answer
392 views

Green's operator for elliptic differential operator

Let $$ P:\Gamma(E)\rightarrow\Gamma(F) $$ be an elliptic partial differential operator, with index zero and closed image of codimension 1, between spaces $\Gamma(E)$ and $\Gamma(F)$ of smooth sections ...
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45 views

Is the spectrum of a first order PDO always unbounded from both sides?

Let $E \to X$ be a smooth vector bundle over a compact Riemannian manifold $X$ and assume that $P:\Gamma(E) \to \Gamma(E)$ is a self-adjoint partial differential operator of order $1$. We think of ...