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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8
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1answer
277 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 ...
8
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1answer
375 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 ...
6
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2answers
2k views

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 ...
6
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0answers
156 views

Fréchet manifold structure on space of sections

I know that the space $\mathsf{C}^\infty(M;N)$ of smooth maps from a closed (smooth) manifold $M$ to a (smooth) manifold $N$ is a Fréchet manifold. I have been looking for a more general version of ...
5
votes
1answer
493 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 ...
5
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1answer
126 views

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 $\...
5
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2answers
52 views

Rigorous global optimization

The work by Thomas Hales (see enter link description here) before the formal proof solves a number of global optimization problems that need to be solved exactly. The strategy relies on following ...
5
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0answers
71 views

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. ...
5
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47 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 ...
4
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1answer
1k 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 \...
4
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1answer
143 views

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 ...
4
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1answer
139 views

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 ...
4
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0answers
108 views

Generalizations of Sard-Smale Theorem

Sard-Smale theorem holds for Fredholm maps $f:M\rightarrow B$ between separable Banach manifolds $M,N$. There are some constrains relating the Fredholm index $\operatorname{ind}(f)$ of $f$ to its ...
3
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1answer
847 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 ...
3
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1answer
154 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 ...
3
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1answer
83 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 $...
3
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1answer
296 views

Few questions about global analysis relating $C^k$ functions

First question is about the definition. Let $U$ be an open subset of $R^n$. Let $f$ be $k$ times continuously differentiable function on $U$. $C^k$ norm of $f$ is defined as sum of uniform norm of $i^{...
3
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2answers
275 views

Vector space structure on $T_pM$ again

This minor problem popped up while I was reading the book "Modern Differential Geometry for Physicists" by Chris J. Isham. It deals with introducing vector space structure on a tangent space $T_pM$ to ...
3
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0answers
126 views

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 $\...
3
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0answers
106 views

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 ...
2
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1answer
125 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 $...
2
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1answer
157 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 ...
2
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1answer
106 views

“Restriction” of a Differential Operator on a Vector Bundle to the Space of Local Sections?

I'm trying to understand the definition of differential operators on vector bundles. The material I'm following http://www.mat.univie.ac.at/~stein/research/talks/nhops.pdf starts with the ...
2
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1answer
55 views

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 ...
2
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0answers
61 views

Defining a general structure of “Calculus” [closed]

I've been thinking lately, is there a way to generalize the fundamental concepts of Calculus such as convergence, differentiability and integrability to it's "maximum potential"? That is, ...
2
votes
1answer
80 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 ...
2
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1answer
254 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: ...
2
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0answers
38 views

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 ...
2
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0answers
239 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 ...
2
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0answers
88 views

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 ...
2
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0answers
68 views

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}\...
1
vote
1answer
30 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 ...
1
vote
1answer
103 views

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 $K\subset \mathbb R^n$ ...
1
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1answer
99 views

Prove that solution blows up in finite time [duplicate]

I want to show that solution of this Cauchy problem \begin{cases} u'(t)=u(t)^2 + t \\ u(0)=0 \end{cases} is defined for $t \in [0,\alpha]$, with $\alpha <3 $ I tried to integrate, but it's not ...
1
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1answer
77 views

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 ...
1
vote
2answers
274 views

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 ...
1
vote
1answer
437 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
35 views

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 ...
1
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1answer
48 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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0answers
27 views

How to show that $\left\{x^*\in X \mid \limsup_{y \to x,\;\|y\|=1}\frac{\langle x^*,y-x\rangle}{\|y-x\|} \le 0\right\} = \mathbb R x$ if $\|x\|=1$

Let $X$ be a Banach space with topological dual $X^\star$. Given an extended value function $f:X \to \mathbb R \cup \{+\infty\}$, defined its Fréchet subdifferential at a point $x \in X$ as $$ \...
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0answers
25 views

How to compute the second fundamental form of the parallel surface [duplicate]

Given a regular surface $S$, we can compute the second fundamental form of it. Now, from the old surface, we can create a new one $T = S + tN$, where $t$ is the given fixed real number and $N$ is the ...
1
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1answer
70 views

Hodge star is conformally invariant on $\Lambda^{n/2}(V)$, for $n$ even

I am studying the Hodge star operator for the first time. I am trying to prove that for $n$ even, then for any $\omega \in \Lambda^{n/2}(V)$ $\star_g \omega= \star_{\tilde{g}} \omega$, where $g$ and $\...
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0answers
65 views

Using the covariant derivative for a Riemannian Metric for $H^1([0,1]; S^2)$

I am currently researching gradient flows on Riemannian Hilbert manifolds and in this paper, Trombe considers the Hilbert space $((H^1([0,1]; \mathbb{R}^3), \langle \cdot, \cdot \rangle_{H})$ with ...
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0answers
45 views

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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0answers
18 views

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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0answers
40 views

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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0answers
24 views

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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0answers
40 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 ...
1
vote
1answer
42 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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0answers
46 views

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 ...