# 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
### 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 ...