Naive thoughts and questions

Riemannian manifolds

• Riemann curvature tensor: We can define this only using the definition of Levi-Civita connection (covariant derivative of a vector field) and the Lie bracket.
  \( \mathrm{Rm}(X,Y)Z = \nabla_X( \nabla_Y Z) - \nabla_Y (\nabla_X Z ) - \nabla_{[X,Y]} Z \)
• We can easily prove the first Bianchi identity from the Jacobi identity for the Lie bracket.
• Ricci tensor and scalar curvature. These are traces. Use local coordinates or orthonormal frames?
  \( R_{jk}=g^{il} R_{ijkl} \) or \( \mathrm{Ric} (X,Y) = \sum_i \mathrm{Rm}(e_i, X) Y \cdot e_i \)
  \( R =g^{jk} R_{jk} \) or \( R = \sum_i \mathrm{Ric} (e_i,e_i) \)
• Decide on sign convention for \( R_{ijkl}\). The one I have chosen is opposite of the standard one.
• Tensors: my guess it is best to consider these, at each point, as multilinear maps.
• Covariant differentiation acting on sections of tensor bundles. This is defined just using the product rule. E.g.,
  Let \( a\) be a 1-form. \( (\nabla_X a) (Y) = X(a(Y)) - a(\nabla_X Y ) \)
  Let \( b\) be a covariant 2-tensor. \( (\nabla_X b) (Y,Z) = X(b(Y,Z)) - b(\nabla_X Y, Z ) - b( Y, \nabla_X Z ) \)
• Exercise: Prove the second Bianchi identity.
• Formula for commutator of covariant derivatives (Ricci formula). Local coordinates? E.g., if \( a\) is a 1-form, then
  \( \nabla_i \nabla_j a_k - \nabla_j \nabla_i a_k = - R_{ijkl} a_l \)
• A general question is: How to calculate? My guess is that local coordinates would be best.
• Rough Laplacian acting on tensors: trace of the second covariant derivative. \( \Delta = g^{ij} \nabla_i \nabla_j\) or \( \Delta = \sum_i \nabla^2_{e_i,e_i} \)
• Aspects of tensor calculus and local coordinate calculations are explained in detail in an appendix of my book with Dan Knopf.
• Bochner formula: \( \tfrac{1}{2} \Delta |\nabla u|^2 = |\nabla^2 u|^2 + \nabla \Delta u \cdot \nabla u + \mathrm{Ric}(\nabla u, \nabla u) \)

Smooth manifolds

• Is the Lie derivative of a tensor defined?
• Differential forms, exterior derivative, and Stokes's theorem? DeRhamCohomology

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