In Paper I of this series we introduced a volume-limited parent sample of 871 galaxies from which we extracted the ATLAS ^ { 3 D }  sample of 260 ETGs .
In Paper II and III we classified the ETGs using their stellar kinematics , in a way that is nearly insensitive to the projection effects , and we separated them into fast and slow rotators .
Here we look at galaxy morphology and note that the edge-on fast rotators generally are lenticular galaxies .
They appear like spiral galaxies with the gas and dust removed , and in some cases are flat ellipticals ( E5 or flatter ) with disky isophotes .
Fast rotators are often barred and span the same full range of bulge fractions as spiral galaxies .
The slow rotators are rounder ( E4 or rounder , except for counter-rotating disks ) and are generally consistent with being genuine , namely spheroidal-like , elliptical galaxies .
We propose a revision to the tuning-fork diagram by Hubble as it gives a misleading description of ETGs by ignoring the large variation in the bulge sizes of fast rotators .
Motivated by the fact that only one third ( 34 % ) of the ellipticals in our sample are slow-rotators , we study for the first time the kinematic morphology-density T - \Sigma relation using fast and slow rotators to replace lenticulars and ellipticals .
We find that our relation is cleaner than using classic morphology .
Slow rotators are nearly absent at the lowest density environments ( f ( SR ) \la 2 \% ) and generally constitute a small fraction ( f ( SR ) \approx 4 \% ) of the total galaxy population in the relatively low density environments explored by our survey , with the exception of the densest core of the Virgo cluster ( f ( SR ) \approx 20 \% ) .
This contrasts with the classic studies that invariably find significant fractions of ( misclassified ) ellipticals down to the lowest environmental densities .
We find a clean log-linear relation between the fraction f ( Sp ) of spiral galaxies and the local galaxy surface density \Sigma _ { 3 } , within a cylinder enclosing the three nearest galaxies .
This holds for nearly four orders of magnitude in the surface density down to \Sigma _ { 3 } \approx 0.01 Mpc ^ { -2 } , with f ( Sp ) decreasing by 10 % per dex in \Sigma _ { 3 } , while f ( FR ) correspondingly increases .
The existence of a smooth kinematic T - \Sigma relation in the field excludes processes related to the cluster environment , like e.g. ram-pressure stripping , as main contributors to the apparent conversion of spirals into fast-rotators in low-density environments .
It shows that the segregation is driven by local effects at the small-group scale .
This is supported by the relation becoming shallower when using a surface density estimator \Sigma _ { 10 } with a cluster scale .
Only at the largest densities in the Virgo core does the f ( Sp ) relation break down and steepens sharply , while the fraction of slow-rotators starts to significantly increase .
This suggests that a different mechanism is at work there , possibly related to the stripping of the gas from spirals by the hot intergalactic medium in the cluster core and the corresponding lack of cold accretion .