Exercise 3.5 (Properties of tangent vectors on manifolds)

Question

extbf{Lemma 3.4 (Properties of Tangent Vectors on Manifolds).} Suppose $M$ is a smooth manifold with or without boundary, $p \in M, v \in T_p M$, and $f, g \in C^{\infty}(M)$.
\begin{enumerate}[label=(\alph*)]
\item If $f$ is a constant function, then $v f=0$.
\item If $f(p)=g(p)=0$, then $v(f g)=0$.
\end{enumerate}

extbf{Exercise 3.5.} Prove Lemma 3.4.

Elu Thingol · Accepted Answer

\begin{enumerate}[label=(\alph*)]
    \item First, consider the special case when $f_1$ is equivalently $1$, i.e., $f(p)=1$ for all $p \in M$. By the product rule (3.4),
    $$v(f_1) = v(f_1 \cdot f_1) = f_1(p) \cdot v(f_1) + f_1(p) \cdot v(f_1) = 2f_1(p) \cdot v(f_1),$$
    which is only possible when $v(f_1) = 0$. In the more general case when $f(p) = c$ for some $c \in \mathbb{R}$, linearity of $v$ gives
    $$v(f) = v(c \cdot f_1) = c \cdot v(f_1) = 0.$$

\item Inserting the values $f(p) =0$ and $g(p)=0$ into the product rule (3.4), we obtain
    \begin{equation*}
        v(fg) = f(p) \cdot v(g) + g(p) \cdot v(f) = 0 \cdot v(g) + 0 \cdot v(f) = 0.
    \end{equation*}
\end{enumerate}

Exercise 3.5 (Properties of tangent vectors on manifolds)

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Exercise 3.5 (Properties of tangent vectors on manifolds)

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