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.
66
questions
3
votes
1answer
847 views
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 ...
2
votes
1answer
157 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 ...
1
vote
1answer
103 views
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 $K\subset \mathbb R^n$ ...
2
votes
0answers
68 views
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
$$\int_{\Omega}\frac{|\nabla u|^2}{|x|^2}d{x} \;\ge\; \left(\frac{N-4}{2}\...
8
votes
1answer
375 views
Reference request: infinite-dimensional manifolds
The following books and/or notes develop various aspects of the theory of infinite-dimensional manifolds:
Lang, Fundamentals of Differential Geometry.
Kriegl & Michor, The Convenient Setting of ...
4
votes
1answer
139 views
When are heat kernels only dependent on the distance?
"All" the examples of heat kernels in circulation are only dependent on the distance between the space variables rather than on the space variables themselves, i.e.
$$K(t;x,y) = K(t;d(x,y)).$$
Think ...
0
votes
1answer
71 views
What's wrong in this prop about volume form if we drop “oriented”?
I was studying Prop 15.29 from Lee's Introduction to Smooth Manifold and I asked myself what's wrong with this proof if we drop the oriented assumption.
I know that I'd came up with a non zero $n$-...
4
votes
1answer
1k views
Hodge star operator and volume form, basic properties
let $(M,g)$ be an oriented Riemannian manifold. Let $*$ be the hodge operator,
I want to prove that $$*\mathrm{vol}_g =1$$ where $\mathrm{vol}_g$ is the associate volume form $\sqrt{g} e^1\wedge \...
5
votes
1answer
493 views
Green's operator for elliptic differential operator
Let
$$
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 ...
5
votes
0answers
47 views
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 ...
1
vote
0answers
156 views
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. ...
2
votes
1answer
106 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 ...
0
votes
0answers
56 views
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 ...
3
votes
2answers
275 views
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 ...
3
votes
1answer
296 views
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^{...
0
votes
1answer
102 views
$C^{0}$-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
