The Riemann-Hurwitz Formula, named for Bernhard Riemann and Adolf Hurwitz, describes the precise relationship between the Euler characteristics of two surfaces when one surface is a ramified covering of the other, accounting for the branch points where the covering map fails to be locally one-to-one. It connects the phenomenon of ramification to algebraic topology, and it serves as a prototype for many later results of the same kind. The formula is a standard tool throughout the theory of Riemann surfaces and algebraic curves, used to compute the genus of a curve from data about a covering map to a simpler one.
Facts
StatementFor a nonconstant holomorphic map between two compact Riemann surfaces that is a ramified covering of degree N, the Euler characteristic of the covering surface equals N times the Euler characteristic of the base surface minus the sum, over every ramification point, of one less than that point's ramification index. 1 Classification
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Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Sources
1. Riemann-Hurwitz formula, Wikipedia
Lead section, first paragraphQuote, Lead section, first paragraph
In mathematics, the Riemann-Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other.
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