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An affine connection $\nabla$ on a differentiable manifold $M$ is a mapping

$$\nabla:\Gamma(M)\times\Gamma(M)\longrightarrow\Gamma(M)$$ denoted $\nabla(X,Y)=\nabla_X Y$ and satisfying

  1. $\nabla_{X+Y}Z=\nabla_X Z+\nabla_Y Z$
  2. $\nabla_{fX}Z=f\nabla_X Z$
  3. $\nabla_X(Z+W)=\nabla_XZ+\nabla_XW$
  4. $\nabla_X(fZ)=f\nabla_XZ+X(f)Z$



The linearity in the first argument with respect to functions of $\Gamma(M)$ denotes a tensorial character, that is, $(\nabla_XY)_p$ depends on $X_p$ but not on the extension of $X$ out of $p$. It is not, however, tensorial in the second argument, stressed in $\nabla_X(fY)=f\nabla_XY+X(f)Y\neq f\nabla_XY$. Thus $(\nabla_XY)_p$ depends on $Y_p$ and on the 'evolution' of $Y$ along the direction $X_p$