  Topological defects are ubiquitous on surfaces with orientational order
fields. Here, we study equilibrium states generated by the feedback between
geometry and nematic order on fluid membranes with an integer topological
defect. When the Frank elastic constants associated with the orientational
field dominate, the surfaces spontaneously deform toward an conical shape
featuring an aster topological defect at its apex. In the case of vanishing
tension this is a solution to the normal force balance. We show that the
stability of the surface depends on the balance of the elastic parameters and
the phase of the defect. When boundary constraints are introduced, we observe
three distinct modes of deformation. These deformation modes take advantage of
the way in which splay, twist and bend distortions of the director field can be
exchanged on a curved surface. We discuss how these deformation modes are
distinguished by their response to the cost of twist distortions and the
existence of inverted solutions. Our findings show that fusion of +1/2
topological defect pairs can reduce the total energy of deformable surfaces.
Finally, we argue how these results can be relevant for biological systems.
