Confusion in the Lagrangian description of Material Surface in Continuum Mechanics

In Lagrangian or Material description, the physical properties are
described in terms of the material coordinates and time. It focuses on
what is occurring at a fixed material point (or particle) labeled by
its material coordinates as time progresses.

My confusion: But the material surface (defined below) is written in terms of material coordinates only in case of the reference configuration and there is no time parameter. I understand that in reference configuration it is logical to not be dependent on time, but in the definition of material description, time is supposed to be included.

In Eulerian or Spatial description, the physical properties are
described in terms of the spatial coordinates and time. It focuses at
a fixed point in space as time progresses.

A material surface is a mobile surface in the space constituted
always by the same particles.

In the reference configuration, the material surface is defined
in terms of the material coordinates as f(X,Y,Z) = 0, where the set
of particles (material points) belonging to the surface are the same
at all times.

In the spatial description, it is defined as f(x,y,z,t) = 0. The
set of spatial points belonging to the surface depend on time, and
the material surface moves in space.