Let P = TensorInnerProduct(g, T, S).  Let g = g_{ij} dx^i &t dx^j and let h = h_{ij} D_x^i &t D_x^j be the inverse contravariant metric.  If  T = t^i*D_x^i and S = s^i*D_x^i are vectors, then P = g_{ij}*t^i*s^j.  If T = t_i*dx^i and S = s_i*dx^i are 1-forms, then P = h^{ij} t_i*s_j.  If T= t_{ij}*dx^i &w dx^j and S= s_{ij}*dx^i &w dx^j are 2-forms, then P = h^{ik}*h^{jl}*t_{ij}*s_{kl}.  If T = t^i_{jk}*D_x^i &t dx^j &t dx^k and S = s^i_{jk}*D_x^i &t dx^j &t dx^k are type [1, 2] tensors, then P = g_{ia}h^{jb}*h^{kc}*t_^i{jk}*s^a{bc} and so on.

This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form TensorInnerProduct(...) only after executing the command with(DifferentialGeometry) and with(Tensor) in that order.  It can always be used in the long form DifferentialGeometry:-Tensor:-TensorInnerProduct.