1547671870-The_Ricci_Flow__Chow

3. THE STRUCTURE OF CURVATURE EVOLUTION 185

We now adopt this general definition to introduce another square of the
operator Rm. Observe that for each x E Mn, the vector space /\^2 TxMn can
be given the structure of a Lie algebra g isomorphic to so (n). Indeed, given
U, V E /\^2 T* Mn, we define their Lie bracket by

(6.25) [U, VLj ~ gke (Uik Vlj - VikUej).

In a local orthonormal frame field { ei}, any 2-form U may be naturally iden-
tified with an antisymmetric matrix (Uij) so that formula (6.25) corresponds
to

[U, VLj = (UV - VU)ij.

This yields an evident Lie algebra isomorphism between g ~ ( /\^2 r; Mn, [., · l)

and so (n) for each x E Mn. We endow g with the inner product defined by

(6.22). Then the construction outlined above applies. In particular, in local

coordinates {xi}, there is a basis { <p( ij) : 1 ::::; i < j ::::; n} for g defined by

<p(ij) ~ dxi /\ dxj.

The corresponding structure constants cfrn),(rs) are defined by

[dxP /\ dxq' dxr /\ dx^8 ] = L cf~)),(rs) dxi /\ dxj.

(ij)

Since

formula (6.25) implies that

Then formula (6.24) yields

(6.26) (R m #) ijk€ -_ R pquv R rswx c(pq),(rs)c(uv),(wx) (ij) (€k) ·

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