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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Tensors in a vector bundle with infinite-dimensional fibers

If $\xi = (\pi,E,M)$ is a vector bundle such that each fiber has finite-dimension, and $V=\Gamma(M,E)$ is the space of smooth sections, then there is an isomorphism between $(\otimes^r V)\otimes(\...
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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 ...
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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 ...
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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 ...
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86 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 ...
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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 $K\subset \mathbb R^n$ ...
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Is the space of smooth sections of a smooth bundle a Fréchet Manifold?

I'm not very prepared on these concepts and i'm wondering if there're some good references addressing this problem... My aim is to present the problem of linearization for the Euler-Lagrange operator ...
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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}\...
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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 ...
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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 ...
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What's wrong in this prop about volume form if we drop “oriented”?

I was studying Prop 15.29 from Lee's Introduction to Smooth Manifold and I asked myself what's wrong with this proof if we drop the oriented assumption. I know that I'd came up with a non zero $n$-...
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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 \...
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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 ...
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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 ...
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Apparently meaningless computation with the Hodge star operator

In the last lecture we started speaking about hodge star operator. Let $E$ be a $n$ dimensional vector space with a non degenerate bilinear form $g$. $\mathcal{O}$ the orientation line of $E$, i.e. ...
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57 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 ...
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Differentiable structure on the gauge group?

In this paper I have come across a formulation involving differentiation in the gauge group of a principal bundle which I do not understand (found at the very top of p. 369). Let $P\rightarrow M$ be ...
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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 ...
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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 $i^{...
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$C^{0}$-norm of a map

I have the following question: Consider a continous map $f:M\rightarrow N$, where $M,N$ are smooth manifolds. How can one define its $C^{0}(M,N)$-norm, i.e. $||f||_{C^{0}(M,N)}$ ? Greetings, Daniel