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

我们表明，假设 Ricci 曲率和注入半径的下限，黎曼流形的某些特征数（包括所有庞特里亚金数和陈数）的绝对值与体积成正比。证明依赖于陈-韦尔理论，该理论应用于由欧几里得连接在图表上构建的连接，其中度量张量是调和的并且具有有界 Hölder 范数。我们将这个定理推广到根据 Hölder 正则性定义的 Gromov-Hausdorff 闭类粗糙黎曼流形。假设有一个额外的 Ricci 曲率上限，我们表明欧拉特性也与体积成正比。此外，我们评论了流形的 Betti 数的体积比较定理，在截面曲率上有一个额外的上限。