Reference for constant sectional curvature manifolds being locally isometric

Question

I am looking for a reference (or a good proof) for the fact that two Riemannian manifolds of same constant sectional curvature are locally isometric.

Answer 1

Riemannian Manifold: An Introduction to Curvature by John M. Lee, proposition 10.10, page 181.

Answer 2

Let $(e_1, \dots, e_m)$ be a local orthonormal frame on $M$, with dual frame $(\omega^1, \dots, \omega^m)$. Let $\omega^i_j = -\omega^j_i$, $1 \le i,j \le m$, be the connection $1$-forms. These satisfy the Maurer-Cartan equations \begin{align*} d\omega^i + \omega^i_j\wedge\omega^j &= 0 \end{align*} If $M$ has constant sectional curvature $K$, then they also satisfy $$ d\omega^i_j + \omega^i_k\wedge\omega^k_j = K\omega^i\wedge\omega^j. $$ Let $(f_1, \dots, f_m)$ be a local orthonormal frame on $N$, with dual frame $(\theta^1, \dots, \theta^m)$ and connection $1$-forms $\theta^i_j$. If $N$ has constant sectional curvature $K$, then these forms also satisfy the equations above.

On $M\times N$, consider the following differential system: \begin{align*} \eta^i &= \omega^i - \theta^i = 0\\ \eta^i_j &= \omega^i_j - \theta^i_j = 0 \end{align*} The Frobenius theorem states that if on $M\times N$, there exist $1$-forms $\alpha^i_j, \alpha^{ik}_j, \beta^{i}_{jk}, \beta^{ik}_{jl}$ such that \begin{align*} d\eta^i &= \alpha^i_j\wedge \eta^j + \alpha^{ik}_j\wedge\eta^j_k\\ d\eta^i_j &= \beta^i_{jk}\wedge\eta^k + \beta^{ik}_{jl}\wedge\eta^l_k, \end{align*} then, given any $(p,q) \in M\times N$, there exists an $m$-dimensional submanifold $\Sigma$ passing through $(p,q)$ on which the differential system holds. $\Sigma$ is necessarily the graph of an map $\Phi: M \rightarrow N$ such that $\Phi^*\theta^i = \omega^i$, $1 \le i \le m$. This implies $\Phi$ is an isometry.

It remains to verify the conditions required by the Frobenius theorem. However, this is straightforward.