# 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.

56 questions
1answer
22 views

### 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 ...
0answers
6 views

### Rank of smooth distribution

Let D is a smooth distribution of vector bundle $\pi: E\rightarrow M$ such that M is a manifold. Why rank of D is lower semi-continuous function on a M?
0answers
18 views

### 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 ...
0answers
25 views

### 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 ...
0answers
40 views

### 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 $$\psi(w) = \int_0^1 | \ \| w(t) \|^2 - 1 | \ dt$$ ...
1answer
73 views

1answer
51 views

### 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 ...
1answer
134 views

### 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 ...
0answers
100 views

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 $F: ... 0answers 34 views ### Generalisation of the Hopf–Rinow theorem Is their a generalisation of the Hopf–Rinow theorem to infinite dimensional manifolds (Frechet, Banach or Hilbert)? I have actually never seen the prove of the Hopf–Rinow theorem so i don't know if ... 1answer 108 views ### Vector-valued forms inside the first jet bundle On page 433 of "Self-duality in four-dimensional Riemannian geometry" by Atiyah, Hitchin and Singer, it is written that$p^*(E \otimes \Lambda^1) \subset p^*J_1(E)$, where$\Lambda^1 \to X$is the ... 0answers 17 views ### What are examples of a second order operational tangent vector on an infinite dimensional Hilbert space. In the book "a convenient setting for global analysis" they describe the order of an operational tangent vector on a convenient vector space. http://www.mat.univie.ac.at/~michor/apbookh-ams.pdf ... 1answer 60 views ### non-negative curvature of space curve I read in most of the textbook that "curvature of a space curve is always non-negative" but I could not understand the intuition behind this that why is so? Give some nice intuition and proof. 1answer 50 views ### Existence of canonical connection on trivial Banach bundle Let$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$the second-factor projection. ... 1answer 43 views ### Diffeomorphisms and their derivatives If a$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 $$\inf_{x\in X}\big|\det [... 0answers 66 views ### Has this functional been studied somewhere? \newcommand{\M}{\mathcal{M}} \newcommand{\N}{\mathcal{N}} \newcommand{\TM}{\operatorname{T\M}} \newcommand{\TN}{\operatorname{T\N}} Let \M,\N be Riemannian manifolds, f:\M \to \N smooth. ... 1answer 78 views ### Why harmonicity is a local property? Given two Riemannian manifolds M,N, we say that f:M \to N is harmonic if it is a critical point of the Dirichlet energy functional. More precisely, this means that for every variation f_t of ... 1answer 309 views ### one parameter family of diffeomorphism \phi_{t} of \mathbb{R}^2 I have the following question: The one parameter family of diffeomorphism \phi_{t} of \mathbb{R}^2 to itself for t\in (\pi,\pi) is defined in polar coordinates (r,\theta) by$$\phi_t(r,\... 1answer 27 views ### 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$connected? I've seen this statement, but have no idea how to use connectedness here. Does anyone ... 1answer 39 views ###$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$. I did not find any special definiton for the above phrase, so I expect an ... 0answers 30 views ### Let a compact group$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 ...
1answer
73 views

### 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?
0answers
100 views

0answers
24 views

### 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 ...
1answer
34 views

### 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 ...
2answers
1k views

### 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 ...
1answer
87 views

1answer
263 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 ...
1answer
99 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 ...
1answer
62 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 $n$-...
1answer
969 views