\begin{align} g_p\left(aU_p+bV_p,Y_p\right) &= ag_p\left(U_p,Y_p\right)+bg_p\left(V_p,Y_p\right) \,, \quad \text{and} \\ g_p\left(Y_p,aU_p+bV_p\right) &= ag_p\left(Y_p,U_p\right)+bg_p\left(Y_p,V_p\right) \,.
A metric tensor on assigns to each point of a metric in the tangent space at in a way that varies smoothly with .