“…• What are the shortest path directions connecting minima? Previous works have excellently characterized these mesophases and phase transitions, including phase diagrams, orientation distribution function profiles, bifurcation analysis, and the use of imaging techniques, calorimetry characterization, and dynamic simulations, accentuating the thermodynamic I-SmA phase transition perspective (Palffy-Muhoray, 1999;Dogic and Fraden, 2001;Mukherjee et al, 2001;Larin, 2004;Urban et al, 2005;Biscari et al, 2007;Das and Mukherjee, 2009;Chahine et al, 2010;Nandi et al, 2012;Izzo and De Oliveira, 2019;Gudimalla et al, 2021;Khan and Mukherjee, 2021;Mukherjee, 2021). Given that the first four key issues are essentially anchored in the spatial features of the energy landscape, we develop, implement, and validate a novel geometric methodology.…”