📂Geometry

Isometries and Local Isometries on Riemann Manifolds

Isometry

Given a Riemannian manifold $(M, g), (N, h)$, a diffeomorphism $f : M \to N$ is called an isometry if the following holds for $f$:

$$\begin{equation} g(u, v)_{p} = h\left( df_{p}(u), df_{p}(v) \right)_{f(p)},\quad \forall p\in M,\quad u,v\in T_{p}M \end{equation}$$

or

$$\left\langle u, v \right\rangle_{p} = \left\langle df_{p}(u), df_{p}(v) \right\rangle_{f(p)},\quad \forall p\in M,\quad u,v\in T_{p}M$$

Here $df_{p} : T_{p}M \to T_{f(p)}N$ is the derivative of $f$.

Local Isometry

Let $(M, g), (N, h)$ be a Riemannian manifold. If the following condition is satisfied, then the differentiable function $f : M \to N$ is called a local isometry at $p \in M$:

There exists a neighborhood $U \subset M$ of $p$ such that $f : U \to f(U)$ satisfies $(1)$.

Moreover, if a local isometry $f : U \to f(U) \subset N$ exists for every point $p$, then Riemannian manifolds $M$ and $N$ are said to be locally isometric.

Immersed Manifold

Let $f : M^{n} \to N^{n+k}$ be an immersion. That is, for all $p \in M$, the derivative $d_{f}p : T_{p}M \to T_{f(p)}N$ of $f$ is injective. If $N$ has a Riemannian metric $h$, then we can consider the following metric on $M$ induced by $f$ $g$:

$$g(u, v)_{p} = h\left( df_{p}(u), df_{p}(v) \right)_{f(p)},\quad u,v \in T_{p}M$$

In this case, $f$ is called an isometry immersion.