Differential Geometry: A diffeomorphism is conformal if and ony if $\langle(df)_p(v_1), (df)_p(v_2) \rangle= \lambda (p) ^{2} \langle v_1,v_2 \rangle$

Question

A diﬀeomorphism $f : S_1 \to S_2$ is conformal if and only if there exists a smooth positive-valued function $\lambda : S_1 \to \mathbb{R}$ such that $$\langle(df)_p(v_1),(df)_p(v_2)\rangle_{f(p)} = \lambda(p)^{2}\langle v_1,v_2\rangle_{(p)}$$ for all $p\in S_1$ and all $v_1,v_2 \in T_pS_1$.

How can we show this if and only if statement, this in the textbok of Differential Geometry of Curves and Surfaces. Are we going to use the definition that "a diffeomorphism is conformal if $\text{d}f$ preserves angles"?

Can you help me?

score 1 · Answer 1 · answered Nov 30 '19 at 14:05

1

Suppose that $f:S_1\rightarrow S_2$ preserves angles. Let $x\in S_1$, $df_x:(T_xS_1,\langle,\rangle_x\rightarrow (T_{f(x)}S_1\langle,\rangle_{f(x)}$ preserves angle, then it is conformal. See the proof in the reference.

Angle preserving linear maps

answered Nov 30 '19 at 14:05

Tsemo Aristide

87,475

Differential Geometry: A diffeomorphism is conformal if and ony if $\langle(df)_p(v_1), (df)_p(v_2) \rangle= \lambda (p) ^{2} \langle v_1,v_2 \rangle$

1 Answers1