Lie sphere geometry

Target entity: Lie sphere geometry
Context triple: [Sophus Lie, hasConceptNamedAfter, Lie sphere geometry]

• A. Theorema Egregium
  Theorema Egregium is Gauss’s celebrated theorem in differential geometry showing that the Gaussian curvature of a surface is an intrinsic property independent of how the surface is embedded in space.
• B. Erlangen Program
  The Erlangen Program is Felix Klein’s influential 1872 framework that classifies and studies geometries based on their underlying symmetry groups and transformation properties.
• C. Cartan connections
  Cartan connections are a geometric framework generalizing affine and Riemannian connections that model curved spaces on homogeneous spaces, developed by Élie Cartan.
• D. Lorentzian geometry
  Lorentzian geometry is the branch of differential geometry that studies manifolds equipped with metrics of Lorentzian signature, providing the mathematical framework for general relativity and spacetime physics.
• E. Nash embedding theorem
  The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Lie sphere geometry
Target entity description: Lie sphere geometry is a branch of differential geometry that studies the properties and transformations of spheres (and related objects like planes and points) using the methods of Lie groups and projective geometry.

• A. Theorema Egregium
  Theorema Egregium is Gauss’s celebrated theorem in differential geometry showing that the Gaussian curvature of a surface is an intrinsic property independent of how the surface is embedded in space.
• B. Erlangen Program
  The Erlangen Program is Felix Klein’s influential 1872 framework that classifies and studies geometries based on their underlying symmetry groups and transformation properties.
• C. Cartan connections
  Cartan connections are a geometric framework generalizing affine and Riemannian connections that model curved spaces on homogeneous spaces, developed by Élie Cartan.
• D. Lorentzian geometry
  Lorentzian geometry is the branch of differential geometry that studies manifolds equipped with metrics of Lorentzian signature, providing the mathematical framework for general relativity and spacetime physics.
• E. Nash embedding theorem
  The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
• F. None of above. chosen