A Riemannian manifold has almost nonnegative sectional curvature if it admits a sequence of Riemannian metric $g_{i}$ such that $sec(g_{i})\geq (-1/i)$ and $D(g_{i})\leq 1$ where $sec(g_{i})$ is the sectional curvature of $g_{i}$ and $D(g_{i})$ is the diameter of $g_{i}$. ( see, https://www.sciencedirect.com/science/article/abs/pii/S0001870820302759)
My questions are those:
1-Could this definition be true for manifold with boundary?
2- What is the definition of the Riemannian curvature of manifold with boundary?
I have not found any things in the literature review about that when the boundary exists.