Global analysis, analysis on manifolds

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• ConceptGlobal analysis

  Global analysis is the study of differential equations, variational problems and analysis on smooth manifolds as whole objects. It looks at properties that depend on the entire manifold and not only on one chart. Its tools include differential operators on vector bundles, spaces of smooth maps, critical point theory and index theory. It links analysis with the topology of manifolds, for instance through Morse theory and the Atiyah–Singer index theorem.

• ConceptGerm of a smooth function

  The germ of a smooth function at a point $p$ of a smooth manifold is the function seen only on arbitrarily small neighbourhoods of $p$. It is the special case of a germ for smooth real functions. The germs at $p$ form a local ring, and this ring carries all local information of smooth calculus at $p$. Tangent vectors, jets and the classification of singularities are all built on germs.

• LemmaHadamard's lemma

  Let $U \subseteq \R^n$ be an open set that is star-shaped with respect to a point $a \in U$. This means that the segment from $a$ to $x$ lies in $U$ for every $x \in U$. Let $f \colon U \to \R$ be a smooth function. Then there are smooth functions $g_1, \dots, g_n \colon U \to \R$ such that $$f(x) = f(a) + \sum_{i=1}^{n} (x_i - a_i)\, g_i(x) \quad \text{for all } x \in U,$$ and $g_i(a) = \dfrac{\partial f}{\partial x_i}(a)$ for $i = 1, \dots, n$, where $\partial f / \partial x_i$ is the partial derivative.

• LemmaBorel's lemma

  Let $(a_n)_{n \ge 0}$ be any sequence of real numbers. Then there is a smooth function $f \colon \R \to \R$ such that $f^{(n)}(0) = a_n$ for every $n \ge 0$.

• ConceptJet

  The $k$-jet of a smooth map at a point records the value and all derivatives up to order $k$ there, and nothing else. In coordinates, it is the Taylor polynomial of degree $k$, but the notion makes sense on smooth manifolds without coordinates. Jets turn differential equations into geometric objects: a differential equation of order $k$ becomes a subset of a space of $k$-jets. They are the basic tool of the jet bundles and of singularity theory.

• ConceptJet bundle

  The $k$-th jet bundle of a fiber bundle $E \to M$ is the manifold of all $k$-jets of local sections of $E$. A point of it records a point of $M$, a value in the fiber and all derivatives up to order $k$ of a section there. It is the natural home of differential equations on bundles: an equation of order $k$ is a subset of the $k$-th jet bundle. Jet bundles are used in the calculus of variations, in the geometric theory of partial differential equations and in field theory.