A connection determined by its action on a frame field

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Problem 3.3.37 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 3.3 "Differential Geometry of Manifolds"

Let GG be a Riemannian manifold with a global frame-field {ei}i=1n\{ e_i\}_{i=1}^n.

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(a)

Show that any connection on GG is completely determined by its effect on the frame field, i.e., by the vector fields ∇eiej\nabla_{e_i}e_j, i,j=1,…,ni,j=1,\ldots ,n.

(b)

Show that when GG is a Lie group with a bi-invariant metric ⟨⋅,⋅⟩\left\langle \cdot , \cdot \right\rangle, and the frame-field is left-invariant, we characterize the Levi-Civita connection on (G,⟨⋅,⋅⟩)(G, \left\langle \cdot , \cdot \right\rangle ) by setting,

∇eiej=12[ei,ej] \nabla _{e_i}e_j = \frac12 [e_i,e_j]

for all i,j=1,…,ni,j=1,\ldots ,n.

References

[CJP+11] Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. Problems and Solutions in Mathematics. 2nd ed., World Scientific, 2011.