Lemma of Morse in dimension 1

Question

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

Answer 1

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 $0$.