Differential Geometry: Isometries preserves angles.

Asked Nov 26 '19 at 07:52

Active Nov 27 '19 at 13:37

Viewed 165 times

The statement is that:

If $f:S_1\to S_2$ is an isometry between two regular surfaces and $v_1, v_2$ are two vectors in $T_pS_1$, then

the angle between $v_1, v_2$ = the angle between $(df)_p(v_1)$ and $(df)_p(v_2)$.

How can we show that by using the first fundamental form?

edited Nov 27 '19 at 13:37

asked Nov 26 '19 at 07:52

mathasker

3

What's the definition of an isometry? – Oliver Jones Nov 26 '19 at 08:56
Let $S_1, S_2$ be two regular surfaces, then diffeomorphism $f:S_1 \ to S_2$ is called isometry if $\forall p \in S, df_p$ preserves the inner product, equivalently, $df_p$ preserves the first fundamental form. – mathasker Nov 26 '19 at 16:46
1

What's the connection between the inner product and angle? – Oliver Jones Nov 26 '19 at 22:10

0 Answers0