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A differentiable map, also known as a differentiable function, is a function between two smooth manifolds that preserves the smooth structure. This means that at every point in the domain, there is a neighborhood where the function can be closely approximated by a differentiable function between Euclidean spaces. In simpler terms, if you pick any point in one space, the function will behave smoothly around that point, allowing you to take derivatives. The important thing here is that differentiability ensures the map is smooth and has a well-defined tangent relationship. This property is crucial when discussing concepts like angle preservation and local conformality, as it allows us to use differential calculus to understand how the map transforms vectors and angles.

Local conformality refers to the property where a map behaves like a scaled isometry in a small neighborhood around any point. This means that angles are preserved and distances are scaled uniformly by a certain factor. Think of it as zooming in on a map function at a particular point: although the global shape might differ, locally, angles remain consistent and every small line segment's length is multiplied by the same factor. In mathematical terms, a map \( \varphi: S_1 \rightarrow S_2 \) is locally conformal at point \( \text{p} \) if there exists a neighborhood around \( \text{p} \) within which the differential \( \text{d\textbackslash varphi\textbackslash _p} \) can be expressed as \( \text{rQ}\f\), where \( \text{r} \) is a nonzero scalar and \( \text{Q} \) is an orthogonal transformation. This fundamental property guarantees that very small shapes and angles are preserved and only scaled.

An orthogonal transformation is a transformation that preserves angles and distances. In the context of linear algebra and geometry, it refers to a linear map that preserves the dot product of vectors. What this means is that if you have two vectors and you transform them using an orthogonal transformation, the angle between the two vectors will remain the same.
Mathematically, a matrix \( \text{Q} \in O(n) \) is said to be orthogonal if \( \text{Q^\text{\textbackslash T}Q} = \text{I} \), where \( \text{I} \) is the identity matrix. Here, \( \text{Q^\text{\textbackslash T}} \) is the transpose of \( \text{Q} \), implying that the columns (and rows) of \( \text{Q} \) are orthonormal vectors.
In the context of angle preservation, when a map's differential \( \text{\textbackslash varphi} \text{d\textbackslash varphi\textbackslash\textunderscore p} \) can be expressed as \( \text{rQ}, \) the transformation \( \text{Q} \) ensures that angles are preserved, illustrating how orthogonal transformations play a crucial role in maintaining geometric properties during transformations.