An isometry of inner product spaces preserves the inner product, therefore, it preserves both lengths and angles. For real inner product spaces, the group of isometries is \(SO(n)\), the special orthogonal group for dimension \(n\), which consists of rotations. These are all of those transformations with determinant equal to one. In a more general sense, for a Riemannian manifold where we have a local way of measuring angles and lengths, via a smoothly varying inner product on the tangent space, we have the notion of an isometry via the differential of a map \(f : M^n \to N^n\):
\begin{equation} g_{ij}(p)dx^i \otimes dx^j(u,v) = h_{ij}(p) dy^i \otimes dy^j (df_p(u), df_p(v)) \tag{1} \label{isometry_condition} \end{equation}where \(g\) is the metric for \(M^n\) and \(h\) is the metric for \(N^n\). Note the use of Einstein summation convention. Suppose we wish to construct an isometry from a neighborhood of a point \(x\) in a Riemannian manifold \(M^n\) (i.e. a dimension \(n\) Riemannian manifold) to \(\mathbb{R}^n\). That is, we wish to solve (foregoing the explicit application of the coefficients of \(g\) and \(h\) to a point \(p\) for brevity):
\begin{equation} g_{ij}dx^i\otimes dx^j = \delta_{ij}dy^i\otimes dy^j, \label{isometry_local} \tag{2} \end{equation}for \(y\). This is a classical problem, and a classical question, of quite clear importance: it tells us when a space is locally like Euclidean space in the sense of angles and lengths. We can do, more or less, normal trigonometry here, rather than dealing with hyperbolic or parabolic geometries. Recalling the Riemannian curvature tensor:
\begin{equation} R(X,Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X,Y]}Z. \label{riemann_tensor} \tag{3} \end{equation}We have the following theorem of Riemann:
In other words, the Riemannian curvature tensor \(\eqref{riemann_tensor}\) is the integrability condition for the system \((\ref{isometry_local})\). There are a lot of ways to prove this theorem, but it's a rather general kind of problem we see emerge in many contexts involving geometry. Evaluating \(\eqref{isometry_local}\) against basis vectors, we can see that there are essentially fewer equations involved:
\begin{equation} \begin{aligned} g_{ij} &= \Sigma dy^k \otimes dy^l(\partial_{x^i}, \partial_{x^j}) \\ &= \Sigma dy^k(\partial_{x^i}) dy^l(\partial_{x^j}) \\ &= \Sigma \frac{\partial y^k}{\partial x^i}\frac{\partial y^l}{\partial x^j} ,\ \ 1 \leq i \leq j \leq n, \end{aligned} \end{equation}where we are solving for \(y^i\) in a neighborhood. This is a PDE, and for example, in dimension \(3\), there would hypothetically be only \(6\) equations suggested by this formulation. However, there is an issue: if we take the exterior derivative of any one of these equations, we gain a potentially new criteria. There are a series of important theorems addressing these kinds of issues, essentially presenting geometric conditions for the integrability of overdetermined systems where, in fact, there are more equations than there may appear, lurking in wait. The first of these theorems is the Frobenius integrability criteria, and one of the more general ones is the Cartan-Kahler theorem. I'll add more to this later.