Riemannian Connection On A Surface - Cartan Structural Equations

Cartan Structural Equations

See also: Curvature form

On the frame bundle E of a surface M there are three canonical 1-forms:

  • The connection form ω, invariant under the structure group K = SO(2)
  • Two tautologous 1-forms θ1 and θ2, transforming according to the basis vectors of the identity representation of K

If π: E M is the nature projection, the 1-forms θ1 and θ2 are defined by

where Y is a vector field on E and e1, e2 are the tangent vectors to M of the orthonormal frame.

These 1-forms satisfy the following structural equations, due in this formulation to Cartan:

(First structural equations)
(Second structural equation)

where K is the Gaussian curvature on M.

Read more about this topic:  Riemannian Connection On A Surface

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