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CAF123
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Homework Statement
Let ##T## be a ##(1, 1)## tensor field, ##\lambda## a covector field and ##X, Y## vector fields. We may define ##\nabla_X T## by requiring the ‘inner’ Leibniz rule, $$\nabla_X[T(\lambda, Y )] = (\nabla_XT)(\lambda, Y ) + T(\nabla_X \lambda, Y ) + T(\lambda, \nabla_X Y ) . $$
(a) Prove that ##\nabla_XT## defines a ##(1, 1)## tensor field.
(b) Prove that the components of the ##(1, 2)## tensor ##\nabla T## are, $$\nabla_cT^{a}_{\,\,\,b} = e_c(T^{a}_{b}) + \Gamma^{a}_{\,\,dc}T^{d}_{b} − \Gamma^{d}_{ bc}T^{a}_{d}$$
(c) Deduce that the Kronecker delta tensor is covariantly constant ##\nabla \delta = 0.##
Homework Equations
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Tensor fields and linearity
The Attempt at a Solution
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Really just need to check what I am going to do is correct. For a), ##\nabla_X T## defines a (1,1) tensor if it is linear in the arguments ##X## and ##Y##? Linear in ##X## by definition of a connection and for ##Y##;
$$\nabla_X (T (\lambda, fY)) = (\nabla_X T)(\lambda fY) + T(\nabla_X \lambda, fY) + T(\lambda, \nabla_X (fY)) $$ which can be written using the axioms of a connection $$ f(\nabla_X T)(\lambda, Y) + fT(\nabla_X \lambda, Y) + fT(\lambda, f \nabla_X Y) + T(\lambda, X(f)Y)$$ It doesn't seem to be linear in Y at all?
Thanks!