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I'm preparing the differential geometric exam, but I don't have the textbook on hand. My teacher's lecture note left a symbol $f^*\langle v,w\rangle _p$ without further explanation, so I don't know what it means. I think it is hard for me to google something especially in the topic of differential geometry--many people have their own definitions and terminologies. So I decide to ask here. Let me first state a definition in our teacher's lecture notes: (the appointed textbook is doCarmo's)

If $S_1,S_2$ are surfaces, a smooth map $f:S_1\rightarrow S_2$ is called local isometry if it takes any curve in $S_1$ to a curve of the same length in $S_2$.

Next, the note said that,

If $f:S_1\rightarrow S_2$ is differentiable, $p\in S_1$ and $v,w\in T_p(S_1)$, then $f^*\langle v,w\rangle _p=\langle df_p(v),df_p(w)\rangle_{f(p)}$, where $f^*\langle\cdot,\cdot\rangle_p$ is a symmetric bilinear form on $T_pS$.

So my question is, what is the symbol $f^*\langle v,w\rangle_p$ here? What does it mean? Is $f^*$ related to the original function $f$?

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    @MichaelHardy OK now I've edited. – 2017-01-01
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    To me it looks like that sentence is the *definition* of the function $f^*\langle\cdot,\cdot\rangle_p$, evidently a function from $T_p(S_1)\times T_p(S_1)$ to $\mathbb R$. – 2017-01-01
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    @Rahul. So that's how it is! I understand soon after your explanation. – 2017-01-01
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    "(easy question)" If this was easy to you you would not ask the question, right? – 2017-01-01
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    @Did I'm sure it is easy to others, it is just a matter of definition and terminology. – 2017-01-01

2 Answers 2

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This is known as the pullback metric. The notation $(f^* g)(v,w) = g(Df(v),Df(w))$ is standard, so this is a fairly natural adaptation to the $\langle \cdot,\cdot \rangle$ notation for the metric.

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I guess it is just a notation for $\langle df_p(v), df_p(w) \rangle$, because it is also used in Elementary Differential Geometry by Andrew Pressley. Quoting (page $127$):

Let $f:S_1 \to S_2$ be a smooth map and let $p \in S_1$. For $v,w \in T_pS_1$, define:

$$f^*\langle v,w\rangle_p = \langle D_pf(v),D_pf(w) \rangle_{f(p)}$$