# Why are biholomorphic maps sometimes called conformal?

According to Wikipedia, and my own observations, in the most general sense, a "conformal map" refers to a map which locally preserves angles. This is a very geometric concept, and thus it is useful to have a word to refer to the geometric intuition.

Also according to Wikipedia, a function is locally angle-preserving if and only if it is holomorphic and its derivative is everywhere non-zero (on its domain). Thus geometrically speaking, it makes sense to call these maps conformal.

However, as I can attest from my own experience, sometimes a different meaning is ascribed to "conformal" in complex analysis, namely meaning biholomorphic. The two definitions are not equivalent; even though being biholomorphic implies being locally angle-preserving, the exponential map is not one-to-one and thus not biholomorphic, despite being holomorphic and having non-zero derivative, thus locally angle-preserving.

Question: Why are biholomorphic maps sometimes called "conformal", when they are only a special case of locally angle-preserving maps of open subsets of $\mathbb{C}$?

This seems like the equivalent of defining "plant" to mean "cactus" and then having no word left over for plants which are not cacti.

• It's just one of those things. Although I would say some progress has been made. When I was studying complex analysis back in the day, the term "biholomorphic" wasn't around much. It is today, and if I were writing a complex analysis text, I would use it in place of "conformal" when appropriate. – zhw. Oct 10 '16 at 21:19