In mathematics, especially in singularity theory, the splitting lemma is a useful result due to René Thom which provides a way of simplifying the local expression of a function usually applied in a neighbourhood of a degenerate critical point.
Formal statement Let f : ( R n , 0 ) → ( R , 0 ) {\displaystyle f:(\mathbb {R} ^{n},0)\to (\mathbb {R} ,0)} be a smooth function germ, with a critical point at 0 (so ( ∂ f / ∂ x i ) ( 0 ) = 0 {\displaystyle (\partial f/\partial x_{i})(0)=0} for i = 1 , … , n {\displaystyle i=1,\dots ,n} ). Let V be a subspace of R n {\displaystyle \mathbb {R} ^{n}} such that the restriction f |V is non-degenerate, and write B for the Hessian matrix of this restriction. Let W be any complementary subspace to V. Then there is a change of coordinates Φ ( x , y ) {\displaystyle \Phi (x,y)} of the form Φ ( x , y ) = ( ϕ ( x , y ) , y ) {\displaystyle \Phi (x,y)=(\phi (x,y),y)} with x ∈ V , y ∈ W {\displaystyle x\in V,y\in W} , and a smooth function h on W such that
f ∘ Φ ( x , y ) = 1 2 x T B x + h ( y ) . {\displaystyle f\circ \Phi (x,y)={\frac {1}{2}}x^{T}Bx+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 functions, to functions invariant under the action of a compact group, ...
References Poston, Tim; Stewart, Ian (1979), Catastrophe Theory and Its Applications, Pitman, ISBN 978-0-273-08429-7. Brocker, Th (1975), Differentiable Germs and Catastrophes, Cambridge University Press, ISBN 978-0-521-20681-5.
