List of Local to Global principles

Question

What are some of the local to global principles in different areas of mathematics?

Answer 1

Differential Geometry

The existence of partitions of unity allows one to transfer local results to global ones.

The Gauss-Bonnet Theorem relates the Gaussian curvature (a local quantity) to the Euler characteristic (a global one).

Answer 2

Diophantine Equations

Hasse Condition: If a Diophantine equation is solvable modulo every prime power (locally) as well as in the reals then it is solvable in the integers.

Hasse Principle is that the Hasse Condition holds for all quadratic Diophantine equations.

Answer 3

Graph Theory

A graph has an Eulerian circuit (global) iff every node has even degree (local).

Answer 4

Number Theory

The "original" (in terms of giving rise to the name) local-global principle for quadratic forms over number fields, due to Hasse, has already been mentioned. Here are two further local-global principles in which Hasse was involved.

1. Two (finite-dimensional) central simple algebras over a number field $K$ are isomorphic if and only if their base extensions to central simple algebras over $K_v$ are isomorphic for every completion $K_v$ of $K$. This is essentially the Albert-Brauer-Hasse-Noether theorem.

2. Class field theory can be formulated both for number fields and for their completions, called respectively global class field theory and local class field theory. (It can formulated also for function fields over finite fields, but let's not worry about that here.) Historically global class field theory came first and the proofs of local class field theory originally depended on global class field theory. Eventually Hasse was able to develop local class field theory in a self-contained way and then use it to prove global class field theory.

Answer 5

Complex Analysis

Analytic continuation might be viewed as a local-to-global principle.

Answer 6

Euler Characteristic

If you count the faces vertices and edges (local information) of a polygonal shape then you can compute the Euler Characteristic which tell you how many holes the shape has (global information).

Answer 7

Riemannian Geometry

Remark: All notatations of curvature used in the following Theorems only depend on the chosen point on the manifold, thus their implications to topological properties makes them classical local $\to$ global statements.

Bonnet-Myers Theorem: Let $(M,g)$ be a complete, connected Riemannian manifold with $\operatorname{Ric}_p(x)\ge \frac{1}{r^2}$ for all $(p,v)\in TM$. Then $M$ is compact and $\operatorname{diam}(M)\le \pi r$

Corollaries:

1. Let $M$ be a complete connected Riemannian manifold with $\operatorname{Ric}_p(x)\ge \delta>0$ for all $(p,v)\in TM$. Then the universal covering $\tilde M$ of $M$ is compact and $\pi_1(M)$ is finite
2. Let $G$ be a connected Lie group which admits a bi-invariant metric and let $\mathfrak{g}=Lie(G)$. Assume that $\cal{Z}(\mathfrak{g})=0$. Then both, $G$ and the universal cover $\tilde G$ are compact and $\pi_1(G)$ is finite.

Weinstein Theorem: Let $(M,g)$ be a compact, connected, oriented Riemannian manifold which has everywhere strictly positive sectional curvature. Further let $f\in \operatorname{Iso}(M)$ be orientation-preserving (-reversing) if $\dim(M)$ is even (odd). Then $f$ has a fixed point

Corollary (Synge Theorem): Let $(M,g)$ be a compact connected Riemannian manifold of dimension $m$ with everywhere strictly positive sectional curvature. Then the following hold:

1. If $m$ is even and $M$ orientable, then $M$ is simply connected
2. If $m$ is odd then $M$ is orientable

Cartan-Hadamard Theorem: Let $(M,g)$ be a complete Riemannian manifold with everywhere non-positive sectional curvature. Then $\exp_p:T_pM\to M$ is a covering map.

Classification Theorem for Manifolds with constant sectional curvature: Let $M$ be a simply connected, complete Riemannanian manifold of constant sectional curvature $k$. Then $M$ is, up to rescaling of the metric, isometric to either $\mathbb{S}^m$, $\mathbb{E}^m$ or $\mathbb{H}^m$.

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References on Riemannian Geometry

M. Berger, A panoramic view of Riemannian Geometry, Springer, 2012.

M. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer, 1999.

M. P. do Carmo, Riemannian Geometry, Birkhäuser, 1992.

S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry, Springer, 2004.

M. Spivak, Differential Geometry, vol. I-II, Publish or Perish, 1979.

J. A. Wolf, Spaces of constant curvature, vol. 372, American Mathematical Society, 2011.

Answer 8

Real analysis

The extreme value theorem can be read as: if a function on a compact set is locally bounded then it's globally bounded. (This is only part of the theorem.) The key word here of course is compact.

Answer 9

And i was studying completely about Hasse's local-Global principle,which has many applications like ,there is a separate fantastic group called the Tate-Shafarevich Group that measures the extent of Failure of Hasse's local-global Principle.it is given by $Ш(E/K)=\mathrm{Ker}(H^1(K,E)\mapsto \prod_{v}H^1(K_v,E))$ which means literally ,"the non-trivial elements of the Tate-Shafarevich group can be thought of as the homogeneous spaces of $A$(where $A$ is an Abelian Variety defined over $K$) that have $K_v$-rational points for every place $v$ of $K$, but no $K$-rational point." and the TS group has many important applications in BSD conjecture,Iwasawa theory....

Answer 10

Complex Analysis

If a $\mathbb C \rightarrow \mathbb C$ function can be differentiated (local) then it can be integrated! (Inspired by Qiaochus answer here).

Answer 11

Number Theory

If an integer $n \not \equiv 0 \pmod m$ (for any $m$) then $n \not = 0$.

Answer 12

Set Theory

The axiom of choice is local, while Global choice is global.

(Global choice, or Axiom E in the NGB set theory, is equivalent to saying that all proper classes are of the same "cardinality", or there is a well-ordering of the universe)

Answer 13

$C^*$-Algebras

Let $\mathfrak{A}$ be a unital $C^*$-algebra, $\mathfrak{C}$ a $C^*$-subalgebra of the center of $\mathfrak{A}$ which contains the unit of $\mathfrak{A}$ and for any maximal ideal $x$ of $\mathfrak{C}$ let $I_x$ be the smallest closed two-sided ideal of $\mathfrak{A}$ containing $x$.

Now the local principle by Allan and Douglas says, that $a\in\mathfrak{A}$ is invertible in $\mathfrak{A}$ if and only if $a+I_x$ is invertible in the quotient algebra $\mathfrak{A}/I_x$ for every maximal ideal $x$ of $\mathfrak{C}$.

One can use this result to tackle questions of invertibility and Fredholmness in operator algebras, e.g. one can characterize Fredholmness properties of Toeplitz operators with piecewise continuous symbols by means of easily checked properties of their respective symbols.

Answer 14

(weak) Nullstellensatz - Commutative Algebra/Algebraic Geometry

Let $I$ be an ideal in $K[x_1, \ldots, x_n]$, where K is an algebraically closed field. If for every point of $K^n$, $I$ contains a polynomial non-vanishing at that point (local), then $I = K[x_1, \ldots, x_n]$ (global).

Answer 15

Calculus of Variations

A path $\gamma$ that satisfies the Euler-Lagrange equation $$\frac{d}{dt}\frac{\partial s}{\partial \dot\gamma} + \frac{\partial s}{\partial \gamma}$$ at all times (a local condition) extremises the action $\int_0^T s(\gamma, \dot \gamma, t)\,dt$ (a global condition).

Consequences:

• Conservation of energy for physical systems obeying Newton's second law

• Isoperimetric inequality

• A soap film that minimizes surface area has zero mean curvature

• A function that is "as smooth as possible" (in the sense of minimizing Dirichlet energy $\langle \nabla f, \nabla f\rangle$) is harmonic

• etc...

Answer 16

Number Theory

Zeta L-functions and Birch-Swinnerton Dyer conjecture: These are quantifies which are defined in terms of local things (like multiplying together a function on primes) and global information (like class numbers) is extracted from them.

Answer 17

• Differential equations and integral equations of all kinds link a function together with it's derivatives (or integrals) which creates a dependency or equation system that connects different local neighborhoods or areas to limit which functions are plausible on a global scope.
• The relation $x<y$ applies "locally" to each pair of numbers, so that we can combine to chains of such pairs to globally build an order or a sorting.

So there's examples from both continuous mathematics and discrete!

Answer 18

Number Theory

If $a$ is a square (or $n$th power) modulo every prime power then, since its $p$-adic valuation is even (or a multiple of $n$) for every $p$, $a$ is a square (or $n$th power).

Unlike the Hasse principle for quadratic forms, this works for any degree.

Answer 19

Analysis

$$\prod_{p}|x|_p = |x|^{-1}$$ gives a way to piece together all local norms to find the value of a global norm.

Answer 20

Game theory and dynamic programming

The one-shot deviation principle states that a strategy profile of a finite extensive-form game is a subgame perfect equilibrium if and only if there exist no profitable one-shot deviations for each subgame and every player.

A more general statement is the principle of optimality from dynamic programming, which is that an optimal policy is such that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision.