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As hyperscalers such as Google, Microsoft, and Amazon play an increasingly important role in today's Internet, they are also capable of manipulating probe packets that traverse their privately owned and operated backbones. As a result, standard traceroute-based measurement techniques are no longer a reliable means for assessing network connectivity in these global-scale cloud provider infrastructures. In response to these developments, we present a new empirical approach for elucidating connectivity in these private backbone networks. Our approach relies on using only lightweight (i.e., simple, easily interpretable, and readily available) measurements, but requires applying heavyweight mathematical techniques for analyzing these measurements. In particular, we describe a new method that uses network latency measurements and relies on concepts from Riemannian geometry (i.e., Ricci curvature) to assess the characteristics of the connectivity fabric of a given network infrastructure. We complement this method with a visualization tool that generates a novel manifold view of a network's delay space. We demonstrate our approach by utilizing latency measurements from available vantage points and virtual machines running in datacenters of three large cloud providers to study different aspects of connectivity in their private backbones and show how our generated manifold views enable us to expose and visualize critical aspects of this connectivity.