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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Gradient of a functional defined on an Hilbert space (with respect to a $W^{1,2}$ inner product)

$\newcommand{\R}{\mathbb R}$ Consider the Hilbert space $X = W^{1,2}(\R)\oplus W^{1,2}(\R)$ (Sobolev spaces). I define a function $F:X\to \R$ as $$F(u,g)= \int_\R u(t)\partial_tg(t) dt.$$ $F$ is ...
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Riemannian submersion with complete fibers is complete

Suppose $M\to N$ is a proper Riemannian submersion where $N$ and every fiber with the induced metric are complete. How to show that $M$ is also a complete Riemannian manifold? I know if the total ...
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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 ...
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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}...
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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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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 ...
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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 ...
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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, ...
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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 ...
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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 $\...
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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 ...
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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 ...
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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 ...
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1 answer
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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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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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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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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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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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9 votes
1 answer
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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 ...
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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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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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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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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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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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3 votes
1 answer
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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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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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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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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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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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2 votes
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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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1 vote
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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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7 votes
2 answers
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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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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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1 vote
1 answer
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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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1 vote
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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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