A framework that classifies and characterizes geometries by their symmetry groups: a geometry is defined as the study of properties invariant under a chosen group of transformations acting on a space, so different choices of transformation groups produce distinct geometries.

Principle

Replace an a priori notion of space by the pair (space, transformation group); invariants under the group determine the meaningful geometric objects and relations, so geometry reduces to group theory plus invariant theory.

Demonstration

Euclidean geometry arises from the group of isometries of R^n (preserving distances); projective geometry arises from the projective linear group acting on projective space; conformal geometry concerns the group preserving angles. Comparing groups explains which notions (distance, angle, incidence) are preserved.

Misapplication

Treating the Erlangen program as prescribing metric information where the chosen symmetry group does not determine it, or conflating global topology with the local symmetry classification, misuses the program; likewise assuming all geometric problems reduce purely to group invariants ignores analytic or local curvature data.

Consequence

Provides a unifying language to compare geometries, to derive invariant constructions, and to classify homogeneous geometries by subgroup structures; it guides model-building in geometry and the search for invariant quantities.

Reversal

A reversal emphasizes local differential invariants (Riemann's program): rather than starting from a global transformation group, one focuses on local metric tensors and curvature invariants that may not be captured by a single global symmetry group.

Boundary

Applies primarily to homogeneous or transformation-based geometries where a distinguished group action is present; it is less prescriptive for problems where local analytic structure, singularities, or combinatorial/geometric-topological invariants dominate.

Semantic Tension

Tension exists between Klein's global symmetry-centered view and Riemann's local metric/curvature perspective; some structures are best understood as group invariants, others require local analysis and differentiable data.

Synthesis

The Erlangen Program organizes geometry by symmetry: pick a transformation group acting on a space and study the structures invariant under it, thereby classifying types of geometry by their admissible symmetries and remaining invariant notions.