  Intrinsic topological superconductors with p-wave pairing are rare in nature.
Its underlying reason is due to the fact that it is usually difficult to change
the relative strength between the singlet and triplet channels for the
electron-electron interaction in material. Here we show that by considering
superconductivity occurring on surfaces of topological insulators (TIs), the
relative strength between the singlet and triplet channels can be changed by
geometry and sizes of TIs. Specifically, we show that pairing of electrons at
different locations on the surface of a topological insulator generally tends
to favor the triplet pairing and can induce topological superconductivity by
controlling the surface curvature and size of the topological insulator. We
illustrate the effects in two configurations, thin film geometry and the
spherical geometry with a sphere or a hemisphere, and find that topological
superconductivity arises with the $p \pm ip$ pairing symmetry dominated in
nanoscale size of the TI. As a consequence, vortices can spontaneously form on
surfaces of topological insulators with roughness of appropriate curvature.
These vortices support a Majorana zero mode inside each core and can be used as
a platform to host Majorana zero modes without invoking real magnetic fields.
Our theoretical discovery opens a new route to realize topological
superconductivity in material.
