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As has already been pointed out, while the tubular radius bounds the curvature globally from below, the curvature information is not enough to correctly estimate this radius. Many examples can be cons …

Let me complement (and compliment?) Igor's answer by providing an explicit definition of bounded geometry from the work of Cheeger and Gromov. A Riemannian manifold $(M,g)$ has $C^k$-bounded geometr …

Let $f:X \to \mathbb{R}$ be a Morse function on some compact submanifold $X \subset \mathbb{R}^n$, and assume that $p \in X$ is not a critical point of $f$. For some $\epsilon > 0$ let $D_\epsilon(p)$ …

Background: Let $X$ be a smooth, compact Riemannian submanifold of euclidean space $\mathbb{R}^n$. H Weyl's tube formula asserts that for sufficiently small $t > 0$, the volume $V(X;t)$ of the radius- …

Let $f,g:\mathbb{R}^m \to \mathbb{R}^n$ be two smooth functions and let $k$ be a strictly positive integer. Write $f \sim_k g$ if at each point in the domain, the determinants of all $k \times k$ mino …

This might be ridiculously obvious, but... For each $n \in \mathbb{N}$, let $M_n$ denote the manifold of $n \times n$ matrices with real entries. It is well known that the $n$-dimensional determinant …