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The main premise of this work is that since large cloud providers can and do manipulate probe packets that traverse their privately owned and operated backbones, standard traceroute-based measurement techniques are no longer a reliable means for assessing network connectivity in large cloud provider infrastructures. In response to these developments, we present a new empirical approach for elucidating private connectivity in today's Internet. Our approach relies on using only "light-weight" ( i.e., simple, easily-interpretable, and readily available) measurements, but requires applying a "heavy-weight" or advanced mathematical analysis. In particular, we describe a new method for assessing the characteristics of network path connectivity that is based on concepts from Riemannian geometry ( i.e., Ricci curvature) and also relies on an array of carefully crafted visualizations ( e.g., a novel manifold view of a network's delay space). We demonstrate our method by utilizing latency measurements from RIPE Atlas anchors and virtual machines running in data centers of three large cloud providers to (i) study different aspects of connectivity in their private backbones and (ii) show how our manifold-based view enables us to expose and visualize critical aspects of this connectivity over different geographic scales.