In mathematics, especially in

singularity theory In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it ...

, the splitting lemma is a useful result due to

René Thom René Frédéric Thom (; 2 September 1923 – 25 October 2002) was a French mathematician, who received the Fields Medal in 1958. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became ...

which provides a way of simplifying the local expression of a

function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...

usually applied in a

neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographically localised community ...

of a degenerate critical point.

Formal statement

Let

f:(\mathbb^n, 0) \to (\mathbb, 0)

be a smooth function germ, with a critical point at 0 (so

(\partial f/\partial x_i)(0) = 0

for

i = 1, \dots, n

). Let ''V'' be a subspace of

\mathbb^n

such that the

restriction Restriction, restrict or restrictor may refer to: Science and technology * restrict, a keyword in the C programming language used in pointer declarations * Restriction enzyme, a type of enzyme that cleaves genetic material Mathematics and log ...

''f'' , _''V'' is

non-degenerate In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space ''V'' is a bilinear form such that the map from ''V'' to ''V''∗ (the dual space of ''V'' ) given by is not an isomorphism. An equivalent definit ...

, and write ''B'' for the

Hessian matrix In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed ...

of this restriction. Let ''W'' be any complementary subspace to ''V''. Then there is a change of coordinates

\Phi(x, y)

of the form

\Phi(x, y) = (\phi(x, y), y)

with

x \in V, y \in W

, and a smooth function ''h'' on ''W'' such that :

f\circ\Phi(x,y) = \frac x^TBx + h(y).

This result is often referred to as the parametrized Morse lemma, which can be seen by viewing ''y'' as the parameter. It is the ''gradient version'' of the implicit function theorem.

Extensions

There are extensions to infinite dimensions, to

complex analytic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex der ...

s, to functions invariant under the

action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...

of a

compact group In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural ge ...

, ...

References

* . * {{citation, first=Th, last=Brocker, title=Differentiable Germs and Catastrophes, publisher=Cambridge University Press, year=1975, ISBN=978-0-521-20681-5. Singularity theory Functions and mappings