Question on tangent spaces

Question

In this question, if I also had that $f$ were a diffeomorphism and $f^k = I$ for some positive integer $k$ would it make a difference to the answer being in the negative? Here, by $f^k$ I mean $f$ composed with itself $k$ time.

Thanks!

Answer 1

If $M$ is a connected smooth manifold and $f$ is a diffeomorphism of $M$ such that $f^k$ is the identity for some $k$, and there exists a point $x \in M$ such that $f(x) = x$ and $df_x$ is the identity, then $f$ is the identity.

Proof: Equip $M$ with any Riemannian metric $g$, and then consider the metric $$ \tilde{g}(v, w) = \sum_{j = 0}^{k-1} g\Big((df)^jv, (df)^jw \Big). $$ The point is that since $f^k$ is the identity, $f$ acts as an isometry on $M$ equipped with the metric $\tilde{g}$.

But for isometries $f$ the condition $f(x) = x$ and $df_x = I$ imply that $f$ is the identity. (One can easily check that the set of points $y$ on which $f(y) = y$ and $df_y = I$ is both open and closed.)