Remarks on the formula for the curvature (Q807990)

Let (M,g) be a Riemannian manifold of dimension n. Let \(w^ 1,...,w^ n\) be a basis of 1-forms on M. Then the Riemannian metric g on M has an expression \(g=g_{jk}w^ j\wedge w^ k\), where \(g=(g_{jk})\) is an \(n\times n\) matrix valued function on M. If \(X,Y\in T_ p(M)\), where \(T_ p(M)\) is the tangent space of M at the point p, then we have the inner product \(<X,Y>_ g\). We set \(\xi =w(X)\), \(\eta =w(Y)\) where w denotes the \({\mathbb{R}}^ n\)-valued 1-form \((w^ 1,...,w^ n)\). We set \((w^ 1,...,w^ n)<\xi,\eta >_ g=<X,Y>_ g.\) We define a linear map \(\beta: {\mathbb{R}}^ n\to Hom({\mathbb{R}}^ n,{\mathbb{R}}^ n)\) by \((\beta (\xi)\eta)^ j=\beta^ j_{kl}\xi^ kn^ l.\) The aim of the present paper is to study the curvature tensor \(R(X,Y)=\nabla_ X\nabla_ Y- \nabla_ Y\nabla_ X-\nabla_{[X,Y]}.\) We denote by \(K(X,Y)_ g=<R(X,Y)Y,X>_ g\) which can be expressed by g,\(\beta\) as follows: \(K(X,Y)_ g=K_ 0(\xi,\eta)_ g+K_ 1(\xi,\eta)_ g+K_ 2(\xi,\eta)_ g\), \[ K_ 0(\xi,\eta)_ g=\frac{1}{4}\| \beta^*(\xi)\eta +\beta^*(\eta)\xi \|^ 2_ g- <\beta^*(\xi)\xi,\beta^*(\eta)\eta >g \] -\(\frac{3}{4}\| \beta (\xi)\eta \|^ 2_ g+\frac{1}{2}<\beta^*(\eta)\xi - \beta^*(\xi)\eta,\beta (\xi)\eta >_ g.\) Here \(\beta^*(\xi)\) denotes the adjoint of \(\beta\) (\(\xi\)) with respect to the metric g of \({\mathbb{R}}^ n\).