# 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 ...
66 views

### 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/...
1 vote
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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 ... 42 views

### 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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1 vote
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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 ...
1 vote
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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, ...
212 views

### 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 ...
1 vote
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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 ...
179 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 ...
1 vote
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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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1 vote
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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 $\phi(x) = \psi ' (x)$. ...
267 views

### 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 ...
1 vote
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 \... 1 vote 0 answers 32 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 ... 1 vote 1 answer 41 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 ... 7 votes 2 answers 3k 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 ... 0 votes 1 answer 98 views ### 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 ... 1 vote 1 answer 53 views ### 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)...
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 ...