In the applied algebraic topology community, the persistent homology induced by the Vietoris-Rips simplicial filtration is a standard method for capturing topological information from metric spaces. In this paper, we consider a different, more geometric way of generating persistent homology of metric spaces which arises by first embedding a given metric space into a larger space and then considering thickenings of the original space inside this ambient metric space. In the course of doing this, we construct an appropriate category for studying this notion of persistent homology and show that, in a category theoretic sense, the standard persistent homology of the Vietoris-Rips filtration is isomorphic to our geometric persistent homology provided that the ambient metric space satisfies a property called injectivity. As an application of this isomorphism result we are able to precisely characterize the type of intervals that appear in the persistence barcodes of the Vietoris-Rips filtration of any compact metric space and also to give succinct proofs of the characterization of the persistent homology of products and metric gluings of metric spaces. Our results also permit proving several bounds on the length of intervals in the Vietoris-Rips barcode by other metric invariants. Finally, as another application, we connect this geometric persistent homology to the notion of filling radius of manifolds introduced by Gromov \cite{G07} and show some consequences related to (1) the homotopy type of the Vietoris-Rips complexes of spheres which follow from work of M.~Katz and (2) characterization (rigidity) results for spheres in terms of their Vietoris-Rips persistence barcodes which follow from work of F.~Wilhelm.

翻译：在应用的代数表层学中,由Veaoris-Rips简易过滤法引发的持久性同系物是从测量空间获取表层信息的标准方法。 在本文中,我们考虑一种不同的、更几何的方法,通过先将给定的测量空间嵌入更大的空间,然后考虑环境测量空间内原始空间的厚度。在这样做的过程中,我们为研究持久性同系物概念构建了一个适当的类别,并表明,在一种分类理论感中,越南-Rips-Rips过滤法的标准持久性同系物对于我们的地球测量持久性持续同系物信息来说是畸形的。我们考虑的是,通过将给给定的测量空间中的维亚-Rips-Rips 填充格的耐久性条纹,同时从一个类别中简单地标度的直系同性共性共性分析结果, 也通过测量空间的直径直径直径直径比比值, 也通过测量结果,在越南的直径直径的直径的直径直径上, 。