Martensitic phase transformations and the shape memory effect have been extensively studied, with various theories proposed to understand their mechanisms. This work develops a model based on differential geometry to distinguish martensitic phase transformations from ordinary phase transformations and shape memory alloys from ordinary martensites, excluding cases involving magnetism.

First, we analyse the surface S of a given charge density distribution in crystals using density functional theory (DFT) for Cu, Al, Na, Zr, and NiTi in different phases.We find that S corresponding to different crystals in different phases converge to different triply periodic minimal surfaces (TPMS). The point group of a crystal is a subgroup of its corresponding TPMS, similar to its relationship with the point group of its corresponding Bravais lattice. Notably, the TPMS of two end phases in a martensitic transformation share the same genus, unlike in ordinary phase transformations.

Next, we explore the topological continuity between the two end phases. Since martensitic transformations are diffusionless, the connections between lattice points and their nearest neighbours remain invariant. These connections can be represented by skeletal graphs derived from TPMS. Hence, the invariance of connections means the topological equivalence between skeletal graphs (TPMS) of the two end phases.

Finally, we investigate the geometric reason that distinguishes shape memory alloys from ordinary martensites. Since shape memory alloys undergo reversible martensitic transformations, we introduce the concept of “geometric entropy,” analogous to thermodynamic entropy but determined by the geometry of S and the arrangement of lattice points. DFT calculations reveal that only shape memory alloys can have zero-barrier transformation paths, which coincide with paths where geometric entropy remains unchanged. Thus, the shape memory effect is possible if a martensitic phase transformation occurs without variation in its geometric entropy.