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Addendum to the paper “on a theorem of Miranda”

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Abstract

An existence theorem is proved for closed convex surfaces whose principal radii of curvature regarded as functions of the unit normal vectorn satisfy the equation

$$R_1 R_2 + \Phi (R_1 + R_2 ,R_1 ,R_2 ,n) + c \cdot n = \varphi (n),$$

wheree is an unknown constant vector being sought along with the surface. This equation has been studied by Miranda, who proved a variety of existence and uniqueness theorems about its solutions. The results obtained lead to a certain simplification in the hypotheses imposed on the functions Φ andϕ in these theorems.

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Literature cited

  1. A. I. Medyanik, “On a theorem of Miranda,” Ukr. Geom. Sb., No. 21, 81–85 (1978).

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  2. C. Miranda, “Su un problema di geometria differentiate in grande per gli ovaloidi,” Ann. Mat. Pura ed Appl.87, 237–269 (1970).

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  3. C. Miranda, “Aggiunde ed errata corrige alla memoria ‘Su un problema di geometria differentiate in grande per gli ovaloidi’,” Ann. Mat. Pura ed Appl.88, 349–355 (1971).

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Authors
  1. A. I. Medyanik
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Additional information

Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 79–82.

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Medyanik, A.I. Addendum to the paper “on a theorem of Miranda”. J Math Sci 48, 63–65 (1990). https://doi.org/10.1007/BF01098046

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  • DOI: https://doi.org/10.1007/BF01098046

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