  We construct a novel Wetterich-type functional renormalization group equation
for gravity which encodes the gravitational degrees of freedom in terms of
gauge-invariant fluctuation fields. Applying a linear-geometric approximation
the structure of the new flow equation is considerably simpler than the
standard Quantum Einstein Gravity construction since only transverse-traceless
and trace part of the metric fluctuations propagate in loops. The geometric
flow reproduces the phase-diagram of the Einstein-Hilbert truncation including
the non-Gaussian fixed point essential for Asymptotic Safety. Extending the
analysis to the polynomial $f(R)$-approximation establishes that this fixed
point comes with similar properties as the one found in metric Quantum Einstein
Gravity; in particular it possesses three UV-relevant directions and is stable
with respect to deformations of the regulator functions by endomorphisms.
