The study of geometric structures and maps that preserve angles (and infinitesimal shapes) up to local scaling, including conformal metrics, Möbius transformations, and conformal invariants such as the conformal class of a metric and the conformal curvature tensors.

Principle

Conformal geometry organizes objects by their angle-preserving properties: the primary equivalence relation is equality up to pointwise positive scalar multiplication of the metric, emphasizing scale-free angle data and invariants under conformal maps.

Demonstration

On a Riemann surface, complex-analytic maps are conformal; the uniformization theorem illustrates conformal classification by showing every simply connected Riemann surface is conformally equivalent to the sphere, plane, or disk.

Misapplication

Confusing conformal equivalence with isometry (metric-preserving) leads to incorrect rigidity claims: two metrics in the same conformal class need not have the same lengths or volumes, only the same angles.

Consequence

Correct use yields tools for solving PDEs invariant under scaling (e.g., Yamabe problem), for studying conformal moduli, and for connecting local curvature data to global topology via conformal invariants.

Reversal

The inverse perspective is scale-preserving (isometric) geometry, where absolute distances and volumes are fixed and angle information alone is insufficient to recover the metric.

Boundary

Focuses on structures defined up to pointwise positive rescaling of a metric or local angle-preserving transformations; excludes problems that demand absolute metric data, rigid isometric classification, or purely combinatorial angle assignments without smooth structure.

Semantic Tension

Tension arises with Riemannian and projective geometries: conformal geometry discards scale but retains angle data, whereas Riemannian geometry keeps full metric data and projective geometry preserves geodesic paths but not angles.

Synthesis

Conformal geometry isolates the angle-preserving core of metric geometry: by passing to conformal classes and studying invariants and PDEs invariant under scaling, it relates local infinitesimal shape to global structure while ignoring absolute size.