Population-Based Structural Health Monitoring (PBSHM), aims to leverage information across populations of structures in order to enhance diagnostics on those with sparse data. The discipline of transfer learning provides the mechanism for this capability. One recent paper in PBSHM proposed a geometrical view in which the structures were represented as graphs in a metric "base space" with their data captured in the "total space" of a vector bundle above the graph space. This view was more suggestive than mathematically rigorous, although it did allow certain useful arguments. One bar to more rigorous analysis was the absence of a meaningful topology on the graph space, and thus no useful notion of continuity. The current paper aims to address this problem, by moving to parametric families of structures in the base space, essentially changing points in the graph space to open balls. This allows the definition of open sets in the fibre space and thus allows continuous variation between fibres. The new ideas motivate a new geometrical mechanism for transfer learning in data are transported from one fibre to an adjacent one; i.e., from one structure to another.
GDP of China is about 11 trillion dollars and GDP of the United States is about 18 trillion dollars. Suppose that we know for the coming years, economy of the US will experience a real growth rate equal to \%3 and economy of China will experience a real growth as of \%6. Now, the question is how long does it take for economy of China to catch the economy of the United States. The early impression is that the desired time is the answer of the equation $11\times1.06^X=18\times1.03^X$. The correct answer however is quite different. GDP is not a simple number and the gap between two countries can not be addressed simply through their sizes. It is rather a geometrical object. Countries pass different paths in the space of production. The gaps between GDP of different countries depend on the path that each country passes through and local metric. To address distance between economies of China and of the US we need to know their utility preferences and the path that China passes to reach the US size. The true gap then can be found if we calculate local metric along this path. It resembles impressions about measurements in the General Theory of Relativity. Path dependency of aggregate indexes is widely discussed in the Index Number Theory. Our aim is to stick to the geometrical view presented in the General Relativity to provide a visual understanding of the matter. We show that different elements in the general relativity have their own counterparts in economics. We claim that national agencies who provide aggregate data resemble falling observers into a curved space time. It is while the World Bank or international organizations are outside observers. The vision provided here, leaves readers with a clear conclusion. If China keeps its growth rate, then the economy of China should catch the economy of the United States sooner than what we expect.