📂Geometry

Differentiable Geometry: Local Isometries

Definition¹

Let’s suppose a function $f : M \to N$ defined between two surfaces is given. If there exists an open set

$$U, V \ \text{ such that }\ p \in U \subset M, V \subset N$$

such that for all points $p \in M$, the contraction mapping $f|_{U} : U \to V$ becomes an isometry, then $M$ and $N$ are said to be locally isometric. Furthermore, such $f$ is called a local isometry.

Description

From the definition of isometry, the strict condition of being surjective is relaxed. Naturally, if $f$ is an isometry, any contraction mapping $f|_{U} : U \to M$ is a local isometry.

From the following theorem, it can be understood that two locally isometric surfaces have the same intrinsic properties at each point. Moreover, isometry defines using curves on the surface, but the following theorem tells us that we can discuss the property of being isometric even without such curves.

Theorem

The following two propositions are equivalent.

• Two surfaces $M$ and $N$ are locally isometric.

• For all $p \in M$, there exist two coordinate patch mappings $\mathbf{x} : U \to M$, $\mathbf{y} : U \to N$ $(p \in \mathbf{x}(U))$ with the same open set $U \subset \mathbb{R}^{2}$ and the same coefficients of the first fundamental form $g_{ij}$.

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