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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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Brownian Motion / heat flow generated by Hodge Laplacian

Let $\square_M = - (dd^* + d^*d)$ be the Hodge Laplacian on the differential forms $\Omega(M)$ (or if you wish, on a fixed $\Omega^k(M)$). What is the stochastic process generated by this operator? ...
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Deformation retract of $C^1$ into $W^{1+s,2}$

I am currently trying to apply Ljusternik-Schnirelman theory to a problem in geometric calculus of variations and have the following problem: I consider an energy functional $\mathcal{E}$ on a ...
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tensors of type $(q,r)$ contra-variant and co-variant

Are the positions of $q$ and $r$ correct in the formula $5.29$ taken from the book by Nakahara: Geomtery Topology and Physics ? I think, that $q$ denotes the covariant part and hence should be $$dx^{\...
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Two different Cheeger set on 2-dimensional sphere

Consider a Riemannian sphere $(S^2,g)$. The Cheeger constant of $(S^2,g)$ is $$ h=\inf_{\gamma} \frac{L(\gamma)}{\min\{S(A_1),S(A_2)\}} $$ where $\gamma$ is closed curve on $S^2$ which divide $S^2$ ...
Enhao Lan's user avatar
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Identifying $H^{-s}(M)$ as the dual space of $H^s(M)$.

Let $M$ be $m$-dimensional compact Riemannian manifold. We choose a local coordinate system $\{(U_i,\phi_i)\}$ such that $U_i$ is diffeomorphic to $\mathbb{R}^m$ and $\overline{U_i}$ is compact and ...
Jun's user avatar
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The Hörmander symbol space $S^{-\infty}(\Omega \times \mathbb{R}^n) \subset S^{m}_{cl}(\Omega \times \mathbb{R}^n)$ is closed

This is Exercise 3.4) in Peter Hintz's Introduction to microlocal analysis Which I am using for exam preparation. Let $\Omega \subset \mathbb{R}^n$ open. Consider for $m \in \mathbb{Z}$ the space $S^m(...
Paul Joh's user avatar
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Quotients of Frechet manifolds

In finite dimensions there is the quotient manifold theorem. Is there a generalization to infinite dimensions? Does anybody know references or maybe standard references for such questions?
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Are $L^2$-closed subspaces of $C^\infty$ finite-dimensional?

I have the following question: Given a closed Riemannian manifold $M$, let $V\subseteq C^\infty(M)$ be a linear subspace that is closed with respect to the $L^2$-norm. The question now is whether $V$ ...
pizzalberto's user avatar
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Compact support of the space of all Fields on a manifold $M$

Let $M^n$ be a smooth Manifold, of dimension $n$. From [Kriegl, Andreas; Michor, Peter W., The convenient setting of global analysis, Mathematical Surveys and Monographs. 53. Providence, RI: American ...
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Dimension of $\mathfrak{X}(M)$ and Diff$(M)$

For a smooth closed Manifold $M^n$(of dimension $n$), if $\mathfrak{X}(M)$ denotes the set of all vector fields on the manifold $M$ then we can say that $\mathfrak{X}(M)$ form a vector space of some ...
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5 votes
2 answers
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Symbol of $d^*$, the adjoint of the exterior derivative $d: \Omega^k(M) \to \Omega^{k+1}(M)$

In my Global Analysis course we are studying the symbols of differential operator. We did the example of the Laplacian $\Delta = dd^* + d^*d$ but there is something I do not really understand. Let me ...
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Partition of Unity in Ismagilov-Morgan-Simon Formula of Localization on complete Riemannian non-compact manifolds

The IMS formula of localization cited in Ch. 3 of Schroedinger Operators with Application to Quantum Mechanics and Global Geometry by H. Cycon, R. Froese, W. Kirsch, and B. Simon depends on the ...
Igor O's user avatar
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Calculate Function global maximum

$f_n: (0, ∞) →ℝ$, $f_n(x) = x/n^2 \cdot e^{-x/n}$ Show that the function $f_n$ has a global maximum with value $1/(ne)$.
hallo hello's user avatar
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Geodesically convexity and convexity in normal coordinates

This question regards an assertion in the proof of Theorem 1.3.1 of Douglas Moore's book "Introduction to Global Analysis, Minimal Surfaces in Riemannian Manifolds" (AMS, 2017). Assumptions/...
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Melnikov's method, homoclinic orbits, and bifurcation values

In nonlinear dynamics, Melnikov's approach provides an intriguing way to detect homoclinic bifurcations and bifurcation values, i.e., the values of the parameter at which a dynamical system exhibits ...
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Sequences in $H^1$ which are orthonormal w.r.t. $L^2$ product

I'm currently reading about Jacobi fields and conjugate points in Riemannian Geometry and Global Analysis by Jost, and a few details are losing me (probably because I have a weak background in ...
infinitylord's user avatar
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Asymptotic expansion of elliptic integrals of 1st and 2nd kind as m approaches 0

I am trying to find a two term asymptotic expansion of the following elliptic integrals of first and second kind as $m\to 0$. $$\int_{0}^{\pi/2} \frac{1}{\sqrt{1-m^2 \sin^2\theta}} d\theta$$ $$\int_{0}...
nchom's user avatar
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Is a a small perturbation of proper map still proper?

Let $X,Y$ be two Banach spaces. Let $f:X\to Y $ be a $C^2$ map. Suppose in addition that $f$ is also proper i.e. $f^{-1}(K)$ is compact for any $K\subset Y$ compact (this is equivalent to $f^{-1}(y)$ ...
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Technical Question: smoothness of $C^p$ $(2\le p<\infty)$ charts in Frobenius Theorem in Lang's Fundamentals of Differential Geometry (1999)

This is a technical question: Lang's Fundamentals of Differential Geometry (1999) has a proof of Frobenius' Theorem for $C^p$ ($2\le p<\infty$) on (possibly, infinite dimensional Banach) manifolds. ...
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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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1 answer
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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 ...
andereBen's user avatar
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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, ...
user405919's user avatar
6 votes
0 answers
311 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 ...
Alec's user avatar
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1 answer
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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 $\...
aras's user avatar
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4 votes
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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 ...
Math-Phys-Cat Group's user avatar
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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 ...
Nhat's user avatar
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5 votes
2 answers
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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 ...
Mathieu Dutour Sikiric's user avatar
2 votes
1 answer
234 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 ...
Siran Victor Li's user avatar
2 votes
1 answer
464 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 ...
J. Salieri's user avatar
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0 answers
274 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 ...
Siran Victor Li's user avatar
1 vote
0 answers
48 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 $$\psi(w) = \int_0^1 | \ \| w(t) \|^2 - 1 | \ dt$$ ...
Nhat's user avatar
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5 votes
1 answer
173 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 $\...
Nhat's user avatar
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3 votes
1 answer
569 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 $...
seeker's user avatar
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2 votes
1 answer
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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 ...
Frieder Jäckel's user avatar
8 votes
1 answer
596 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 ...
Oliver Watt's user avatar
2 votes
1 answer
655 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: ...
Timo Dimi's user avatar
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0 answers
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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 ...
Netivolu's user avatar
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4 votes
1 answer
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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 ...
Augusto Pereira's user avatar
1 vote
0 answers
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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 ...
Netivolu's user avatar
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0 votes
1 answer
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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.
Umar Khan's user avatar
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1 answer
174 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. ...
H1ghfiv3's user avatar
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0 votes
1 answer
148 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 [...
Xuxu's user avatar
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7 votes
0 answers
118 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. ...
Asaf Shachar's user avatar
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3 votes
1 answer
123 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 $...
Asaf Shachar's user avatar
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1 vote
1 answer
873 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,\...
Tensor_Product's user avatar
1 vote
1 answer
34 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 ...
MPB94's user avatar
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0 votes
1 answer
66 views

$G$ 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 ...
MPB94's user avatar
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2 votes
0 answers
55 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 ...
MPB94's user avatar
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0 votes
1 answer
230 views

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?
MPB94's user avatar
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3 votes
0 answers
179 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 $\...
Mohan Swaminathan's user avatar