Different definitions of Riemann tensor components Hobson (General Relativity 2006 ed. page 158 eq. 7.13) and Ryder (Introduction to General Relativity 2009 ed. page 124 eq. 4.31) define Riemann tensor components as
$$ {R^\alpha}_{\beta\gamma\delta} = \partial_\gamma {\Gamma^\alpha}_{\beta\delta} - \partial_\delta {\Gamma^\alpha}_{\beta\gamma} + {\Gamma^\alpha}_{\epsilon\gamma} {\Gamma^\epsilon}_{\beta\delta} - {\Gamma^\alpha}_{\epsilon\delta} {\Gamma^\epsilon}_{\beta\gamma} $$
while Weinberg (Gravitation and Cosmology 1972 ed. page 133 eq. 6.1.5) defines
$$ - {R^\alpha}_{\beta\gamma\delta} = \partial_\gamma {\Gamma^\alpha}_{\beta\delta} - \partial_\delta {\Gamma^\alpha}_{\beta\gamma} - {\Gamma^\alpha}_{\epsilon\gamma} {\Gamma^\epsilon}_{\beta\delta} + {\Gamma^\alpha}_{\epsilon\delta} {\Gamma^\epsilon}_{\beta\gamma} $$
Notice the last two terms sign changes completely so the two definitions seem incompatible: why these two different definitions?
 A: There are different conventions for the signature (number of positive, negative and zero eigenvalues counted with multiplicity for a symmetric square matrix, according to Sylvester's law of inertia, those values are independent of the basis used) of the Minkowski metric, called the west coast and east coast convention with signature $(1,3,0)$ and $(3,1,0)$ respectivly:
$$\eta_{\mu\nu}=\operatorname{diag}(+1,-1,-1,-1)$$
$$\eta_{\mu\nu}=\operatorname{diag}(-1,+1,+1,+1)$$
The first one was used by Richard Feynman (living at the West Coast of the US), while the second one was used by Julian Schwinger (living at the East Coast of the US). (By the way, they probably had one or another heated discussion about the conventions as they got a Nobel Prize together.) There are many reasons for the existence of both convetions, one convention is better for doing special or general relativity while he other is better for doing quantum field theory.
All of this effects in particular general relativity: Note, that when you change the sign of the metric tensor (as well as its inverse indirectly) the sign in front of the Christoffel symbol:
$$\Gamma_{\mu\nu}^\sigma
=\frac{1}{2}g^{\sigma\kappa}\left(\partial_\mu g_{\kappa\nu}+\partial_\nu g_{\kappa\mu}-\partial_\kappa g_{\mu\nu}\right)$$
won't change and therefore the entire Riemann tensor won't as well. Therefore I assume, that when you change the convention, you also change the minus sign in the definition of the Christoffel symbol, which would explain, why the first two terms in the Riemann curvature tensor (with an odd number of them) are affected while the last two terms (with an even number of them) aren't.
Different definitions making all of this worse are for example done for the Ricci tensor (contraction of the Riemann curvature tensor), but the two different definitions $R_{\mu\nu}=R_{\mu\sigma\nu}^\sigma$ and $R_{\mu\nu}=R_{\mu\nu\sigma}^\sigma$ again differ my a minus sign due to the algebraic symmetries of the Riemann curvature tensor.
A little overview about this chaos is also given here.
