A partial differential equation, the local M\"obius equation, is introduced in Riemannian geometry which completely characterizes the local twisted product structure of a Riemannian manifold. Also the characterizations of warped product and product structures of Riemannian manifolds are made by the local M\"obius equation and an additional partial differential equation.

Keywords:

submersion, Möbius equation, twisted product, warped product, product Riemannian manifolds submersion, Möbius equation, twisted product, warped product, product Riemannian manifolds

MSC Classifications:

53C12, 58J99 show english descriptions Foliations (differential geometric aspects) [See also 57R30, 57R32]
None of the above, but in this section 53C12 - Foliations (differential geometric aspects) [See also 57R30, 57R32]
58J99 - None of the above, but in this section

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