  This paper studies pseudo-bulges (P-bulges) and classical bulges (C-bulges)
in Sloan Digital Sky Survey central galaxies using the new bulge indicator
$\Delta\Sigma_1$, which measures relative central stellar-mass surface density
within 1 kpc. We compare $\Delta\Sigma_1$ to the established bulge-type
indicator $\Delta\langle\mu_e\rangle$ from Gadotti (2009) and show that
classifying by $\Delta\Sigma_1$ agrees well with $\Delta\langle\mu_e\rangle$.
$\Delta\Sigma_1$ requires no bulge-disk decomposition and can be measured on
SDSS images out to $z = 0.07$. Bulge types using it are mapped onto twenty
different structural and stellar-population properties for 12,000 SDSS central
galaxies with masses 10.0 < log $M_*$/$M_{\odot}$ < 10.4. New trends emerge
from this large sample. Structural parameters show fairly linear log-log
relations vs. $\Delta\Sigma_1$ and $\Delta\langle\mu_e\rangle$ with only
moderate scatter, while stellar-population parameters show a highly non-linear
"elbow" in which specific star-formation rate remains roughly flat with
increasing central density and then falls rapidly at the elbow, where galaxies
begin to quench. P-bulges occupy the low-density end of the horizontal arm of
the elbow and are universally star-forming, while C-bulges occupy the elbow and
the vertical branch and exhibit a wide range of star-formation rates at fixed
density. The non-linear relation between central density and star-formation
rate has been seen before, but this mapping onto bulge class is new. The wide
range of star-formation rates in C-bulges helps to explain why bulge
classifications using different parameters have sometimes disagreed in the
past. The elbow-shaped relation between density and stellar indices suggests
that central structure and stellar-populations evolve at different rates as
galaxies begin to quench.
