morphology dependence of the TF slope (Springob et al. 2007, Masters et al. 2008). Such feature is also predicted by hierarchical galaxy formation models (Tonini et al. 2011), which naturally produce a varying TF slope that depends on the galaxy internal dynamics, a signature of its assembly history. Indeed, there is an extensive body of work in the literature supporting the observational evidence of a dependence of the TF on galaxy type; the TF for late-type spirals (Masters et al. 2006, Courteau 1997, Giovanelli et al. 1997) is not followed by early-type spirals and S0 types (Williams et al. 2010, Bedregal et al. 2006, Neistein et al. 1999), dwarf galaxies (McGaugh et al. 2000, Begum et al. 2008), barred spirals (Courteau et al. 2003), and polar ring galaxies (Iodice et al. 2003).

The evidence suggests that a third parameter beside circular velocity and luminosity enters the TF relation. This parameter depends on galaxy type, and expands the TF into a three-dimensional manifold that describes the structure of spirals or alternatively, of all galaxies. A general three-dimensional manifold for galaxies regardless of type has been investigated (see Zaritsky et al. 2008), with the set of observables ($V_c, I_e, r_e$) in analogy 