Holonomy Analysis of Optical-polarization Temperature Trajectories in Stress-induced Ferroelectric SrTiO$_3$
We develop a data-induced geometric framework for temperature trajectories in optical-polarization data and apply it to temperature-dependent birefringence imaging of stress-induced ferroelectric SrTiO3. By treating the measured response as an observable projection of local material states, the optical-polarization response at each pixel is cast as a trajectory along the temperature axis. Singular value decomposition separates trajectory-induced geometric objects associated with optical-polarization-direction structure and temperature-evolution modes. Holonomies are then defined by transporting these objects around closed loops in real space. The resulting maps reveal spatially localized connection mismatches correlated with an enhanced ferroelectric transition temperature and stress-related optical anisotropy. Local order parameters and angular-gradient analyses confirm that these loop-level signals are distinct from orientational disorder and simple spatial variation. The signed holonomy of the temperature-evolution modes further resolves positive and negative connection structures under a fixed frame convention. These results demonstrate that the data-induced connection geometry of temperature trajectories provides an experimentally accessible diagnostic of electromechanical inhomogeneity in SrTiO3 under stress, without explicitly reconstructing hidden strain or electric-polarization fields.