The Cartan Structural Equations in Classical Riemannian Geometry

There seem to be two ways to approach Riemannian geometry, one thru the Calculus of Variations, the other thru the theory of connexions. These two approaches also tend in different directions within Riemannian geometry, the Calculus of Variations approach emphasizing (before one gets to Morse theory) metric properties, the theory of connexions emphasizing group theoretic aspects.