Part III : Differential geometry ( Michaelmas 2003 , 2004 , 2014 )

Positive-definite means that gp(v, v) > 0 for every v 6= 0, v ∈ TpM . Smoothly depending on p means that for every pair Xp,Yp of C ∞-smooth vector fields on M the expression gp(Xp, Yp) defines a C ∞-smooth function of p ∈M . Alternatively, consider a coordinate neighbourhood on M containing p and let x, i = 1, . . . , dimM be the local coordinates. Then any two tangent vectors u, v ∈ TpM may be written as u = u( ∂ ∂xi )p, v = v ( ∂ ∂xi )p and gp(u, v) = gij(p)u v, where the functions gij(p) = g( ( ∂ ∂xi )