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If we have a vector fiber bundle with a connection $D_X=D(X)$ and an endomorphism $e$; we can then define a new tensor by the following formula: $$E(e)=D_X e D_Y + D_Y e D_X - e D_Y D_X - D_X D_Y e - …

Let $(M,g)$ be a riemannian manifold and $R(X,Y)$ the riemannian curvature as a two form with values in the endomorphisms of the tangent bundle. I define: $$ D_g(X,Y)=det(R)(X,Y) $$ with $det$ the det …

If we consider a spin manifold $M$, we can define the Ricci curvature $Ricc (X,Y)$ which is a symmetric tensor, moreover the spinors are defined, so that we can define a Dirac-Ricci operator: $$DR(\ps …

Let us define a Spin-H structure as a reduction of a SO(n)-bundle by the group: $$Spin^H (n)=Spin(n) \times SU(2)/\{ 1,-1\}$$ The Spin-H structures are analogous to the well-known Spin-C structures bu …

I take an hermitian manifold $(M,g,J)$ and I define from the riemannian curvature $R(X,Y)$ as a $2$-form: $$ Ricc(J)= \sum_i R(J e_i,e_i) $$ with $(e_i)$ an orthonormal basis of the tangent. $$ 2R(J)= …