  Alternative gravity theories have been put forward to address the limitations
of general relativity, each exhibiting differences, particularly in their
strong-field limits. Nonetheless, there remains a lack of effective means to
distinguish or test these theories through local strong-field measurements. In
this work, we define a global Gaussian bending measure over singular spacetime
regions, establish a corresponding global theory, and demonstrate its
applications in a general stationary spacetime. The global theory is based on
differential geometry, rather than on specific gravity theories, allowing it to
depict various physics within general relativity and beyond. For example, it is
applicable to describe the gravitational bending of massless or massive
messengers, such as photons, neutrinos, cosmic rays, and possibly massive
gravitational waves predicted in certain theories of gravity. Besides, the
global theory can be applied to any stationary spacetime regions outside a
rotating black hole. As an instance of its direct applications, we investigate
the highly-curved spacetime effects of the black hole in its immediate
surrounding regions and design local strong-field experiments involving
different shapes of singular lensing patches. New means can be therefore
anticipated to be developed according to the global theory to differentiate
between different gravity theories and test them in their strong-field regions.
