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### Contraction of the second Bianchi identity

The second Bianchi identity is $${R^a}_{b[cd;e]}=0$$ And contracting it with respect to $a$ and $e$ we get $${R^a}_{b[cd;a]}=0 \Leftrightarrow$$ $${R^a}_{bcd;a}+R_{bc;d}-R_{bd;c}=0$$ What I don't ...
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Can somebody please explain to me how the following statement is true? The Riemann curvature tensor $R^c_{dab}$ is given by the Ricci identity (\nabla_a\nabla_b-\nabla_b\nabla_a)V^c\equiv ... 0answers 184 views ### Riemannian curvature and its application on covariant derivative of tensors This identity can be generalized to get the commutators for two covariant derivatives of arbitrary tensors as follows  \begin{align} T^{\alpha_1 \cdots \alpha_r}{}_{\beta_1 \cdots \beta_s ; ... 0answers 317 views ### How to derive covariant derivative and Lie derivative of tensors 1) As title says, how does one derive the following equation for covariant derivate of tensor: A^{\alpha}{}_{;\beta} = A^{\alpha}{}_{,\beta} + \Gamma^{\alpha} {}_{\gamma\beta}A^\gamma where ... 2answers 150 views ### Tensor Components I would like to ask something On the Barrett Oneill's Semi-Riemann Geometry there is a definition of tensor component: Let \xi=(x^1,\dots ,x^n) be a coordinate system on \upsilon\subset M. If A ... 0answers 168 views ### What are spinor fields? For example, a tensor field T=T_{a,b}^{\ \ \ \ c} on a manifold M should be thought as T_{a,b}^{\ \ \ \ c}dx^{a}\otimes dx^{b}\otimes \frac{\partial}{\partial x^{c}} $$with local coordinate ... 1answer 56 views ### Question on tensor calculation in Reimannian geometry Given a Riemannian manfiold M with metric g=(g_{i,j}). Let T=T_{A,B,C,\dots}^{a,b,c,\dots} be a tensor on M. I would like to compute for example T_{A,B,C,\dots}^{a,n,c,\dots}g_{n,m}. We ... 2answers 284 views ### Is there such a thing as discrete Riemannian geometry? General relativity expresses gravity as a curvature in space time, created by the stress energy tensor.$$T_{\mu\nu} \approx R_{\mu\nu} - \frac{R}{2} g_{\mu\nu} Given I put the fact that energy is ...
If $M$ is a s-R(semi-Riemannian) submanifold of a s-R manifold $\overline{M}$ the function $R^{\perp}:\mathfrak{X}(M)\times\mathfrak{X}(M)\times\mathfrak{X}(M)^\perp\rightarrow\mathfrak{X}(M)^\perp$ ...