A volume comparison theorem for characteristic numbers

We show that assuming lower bounds on the Ricci curvature and injectivity radius absolute value of certain characteristic numbers a Riemannian manifold, including all Pontryagin Chern numbers, is bounded proportionally to volume. The proof relies Chern–Weil theory applied connection constructed from Euclidean connections charts in which metric tensor harmonic has Hölder norm. generalize this theorem Gromov–Hausdorff closed class rough manifolds defined terms regularity. Assuming an additional upper bound, we also Euler Additionally, remark volume comparison for Betti with bound sectional curvature. It consequence result by Bowen.