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\(\nabla\)-flat functions on manifolds. (English) Zbl 1098.53013

Let \(M\) be a connected real analytic manifold endowed with an affine connection \(\nabla\). The author proves that a smooth function that is pointwise \(\nabla\)-flat is real analytic and \(\nabla\)-flat. Also, he shows that the ring of all \(\nabla\)-flat functions on \(M\) is an integral domain. Finally, when \(M\) is a complete Riemannian manifold and \(\nabla\) is the Levi-Civita connection on \(M\), he proves that any \(\nabla\)-flat and bounded function on \(M\) is a constant.

MSC:

53B05 Linear and affine connections
53B20 Local Riemannian geometry
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