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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.

Global analysis uses techniques in infinite-dimensional manifold theory and topological spaces of mappings to classify behaviors of differential equations, particularly nonlinear differential equations. These spaces can include singularities and hence catastrophe theory is a part of global analysis. Optimization problems, such as finding geodesics on Riemannian manifolds, can be solved using differential equations so that the calculus of variations overlaps with global analysis

82 questions
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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 Global Analysis. Choquet-Bruhat & DeWitt-Morette,…
mdg
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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 spectrum' is usually meant in this context? The…
Oliver Watt
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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. Let $\bigwedge^k df:\Lambda_k(\TM) \to…
Asaf Shachar
  • 25,967
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2 answers

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 fields form a $1-$D vector space generated by $X =…
Dark_Knight
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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 it's starting point. (But this is too crude for…
Asaf Shachar
  • 25,967
6
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0 answers

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 this statement along the following lines: Let $p:…
Alec
  • 333
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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 $\sqrt{g} e^1\wedge \cdots \wedge e^n$ In these notes it's given as a…
Luigi M
  • 4,053
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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 explain. For $d: \Omega^k(M) \to \Omega^{k+1}(M)$…
Falcon
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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 strategy: Use of interval arithmetic in order to be…
Mathieu Dutour Sikiric
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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 $\langle u,v \rangle_1 = \langle u,v \rangle_0 +…
Nhat
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2 answers

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 to the heat kernel on the Euclidean…
Giovanni De Gaetano
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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 of vector bundles $E\rightarrow M$ and…
mdg
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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 this operator as an unbounded operator $P:L^2(E) \to…
Meneldur
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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 $V$ is finite-dimensional. My idea is to use…
pizzalberto
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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 differentiablity class. More precisely, we need to…
Math-Phys-Cat Group
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