Geodesic
The problem of local rigidity in the class $C^\infty$
Matematičeskie zametki, Tome 22 (1977) no. 5, pp. 643-648
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For a surface of positive curvature with an isolated point of flattening, the possibility of local rigidity is established by means of studying the asymptotic behavior of the field of an infinitesimal deformation in a neighborhood of the flat point and then applying the apparatus of the theory of generalized Cauchy–Riemann systems with singular coefficients. 
@article{MZM_1977_22_5_a4,
     author = {Z. D. Usmanov},
     title = {The problem of local rigidity in the class $C^\infty$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {643--648},
     year = {1977},
     volume = {22},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1977_22_5_a4/}
}
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Z. D. Usmanov. The problem of local rigidity in the class $C^\infty$. Matematičeskie zametki, Tome 22 (1977) no. 5, pp. 643-648. http://geodesic.mathdoc.fr/item/MZM_1977_22_5_a4/