Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I want to write the Morse lemma which is in dimension $n$ :

Let $p$ be a non-degenerate critical point for $f$.

Then there is a local coordinate system $(y^1,...,y^n)$ in a neighborhood $U$ of $p$ with $y^i(p) = 0$ for all $i$ and such that the identity $f= f(p) - (y^1)^2- ... -(y^{\lambda})^ 2 + (y^{\lambda +1})^2+...+(y^n)^2$ holds throughout $U$, where $\lambda$ is the index of $f$ at $p$.

into dimension 1 .

but i don't know how ? , beacause i don't know hwo are the $y^i$ functions ?

please , hel me

thank you

share|cite|improve this question

The $y^i$ are just the local coordinates. For dimension $1$, just take $n = 1$:

Let $M$ be a smooth $1$-manifold and $f: M \longrightarrow \Bbb R$ be a smooth function. Suppose $p$ is a non-degenerate critical point of $f$.

Then there exists a local coordinate system $(y^1)$ in a neighborhood $U \subset M$ of $p$ with $y^1(p) = 0$ satisfying the identity $$f(y^1) = f(p) - (y^1)^2$$ if the Morse index of $f$ at $p$ is $1$ and the identity $$f(y^1) = f(p) + (y^1)^2$$ if the Morse index of $f$ at $p$ is $0$.

share|cite|improve this answer
ok, thank you, please a local coordinate systeme is an Atlas ? – Vrouvrou Apr 2 '13 at 17:45
@Vrouvrou: An atlas is a collection of local charts $(U_\alpha, \varphi_\alpha)$. So each $\varphi_\alpha$ is a homeomorphism $\varphi_\alpha: U_\alpha \longrightarrow \Bbb R^n$. So we can write $\varphi_\alpha$ in terms of its components $y^i$ in $\Bbb R^n$: $\varphi_\alpha(p) = (y^1(p), \dots, y^n(p))$. The $y^i$ are called the local coordinates for the chart $(U_\alpha, \varphi_\alpha)$. Note that $f(y^1, \dots, y^n)$ is really shorthand for $(f \circ \varphi_\alpha^{-1})(y^1, \dots, y^n)$, since $f$ is a function on $M$ but the $y^i$ map to $\Bbb R$. – Henry T. Horton Apr 2 '13 at 18:00
so ,i can say: Let f : M→R a Morse function and x ∈ M a critical point of f. then there existe a charte of M, $\psi : U\rightarrow R$, with $\psi(x) = 0$, such that $f \circ \psi^{−1} : R\rightarrow R$ soit de la forme $f \circ \psi^{−1}(x) = −x^2$ if index of $f$=1 , and $f \circ \psi^{−1}(x) = x^2$ if the index of $f=0$ – Vrouvrou Apr 2 '13 at 20:03
@Vrouvrou: That's almost right, but I would call the critical point something else, like $p \in M$, so that $\psi(p) = 0$. This is because $p$ is in the manifold but $x$ is in $\Bbb R$. – Henry T. Horton Apr 2 '13 at 20:31
ok ,so i replace x by p ! – Vrouvrou Apr 2 '13 at 20:42

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.