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This follows simply from the fact that levi civita connection is defined in such a way that the energy integrals over geodesics be stationary.

This follows simply from the fact that levi civita connection is defined in such a way that the energy integrals over geodesics be stationary. Hamiltionian vector fields are time flows of sympletic manifolds given by the principle of extreme action. If your action consists only of the hamiltonian in question, extremizing it will give geodesics as integral curves of the hamiltonian vector field.

This follows simply from the fact that levi civita connection is defined in such a way that the energy integrals over geodesics be stationary. To extremise the lenght of a curve between two points is the same as to extremise the action consisting only of the energy function, both of which will give geodesics as their integral curves.

This follows simply from the fact that levi civita connection is defined in such a way that the energy integrals over geodesics be stationary.

This follows simply from the very construction of levi civita connection is such that the energy integrals over geodesics are stationary. To extremise the lenght of a curve between two points is the same thing as to extremise the action consisting only of the energy function on the tangent bundle, both of which will give geodesics as their integral curves.

This follows simply from the fact that levi civita connection is defined in such a way that the energy integrals over geodesics be stationary. To extremise the lenght of a curve between two points is the same as to extremise the action consisting only of the energy function, both of which will give geodesics as their integral curves.