- #1

Kreizhn

- 743

- 1

So the statement (if you will allow me to use differential forms) is

Let M be a smooth manifold. Suppose [itex] dg_i [/itex] are linearly independent at each point [itex] p \in M [/itex]. If p is a local extremum of f restricted to M then [itex] \exists \lambda_1,\ldots,\lambda_k \in \mathbb R [/itex] such that

[tex] df(p) = \lambda_1 dg_1(p) + \cdots + \lambda_k dg_k(p) [/tex]

Now my issue is that it's well-known that this should be

[tex] \lambda_0 df(p) = \lambda_1 dg_1(p) + \cdots + \lambda_k dg_k(p) [/tex]

Now normally of course, this wouldn't matter. We could just "normalize by [itex]\lambda_0[/itex]" to get rid of it, unless of course [itex] \lambda_0 = 0 [/itex]. I've seen most of the proofs of this theorem and the [itex] \lambda_0 [/itex] never arises. I can't convince myself why it should indeed arise, yet I can give a great deal of evidence that it is necessary.

Am I confusing this with another LM theorem? Does the [itex]\lambda_0 [/itex] only arise when we consider problems on sub-Riemannian and symplectic manifolds? In those cases, the [itex] \lambda_i [/itex] are not elements of [itex] \mathbb R [/itex] but instead are covectors in [itex] T^*_p M [/itex].