A function with only Morse extrema and critical circles on a cylinder, torus, disk, or sphere always factors as a smooth value-reparametrization of the standard height, radius, or projection function, up to a diffeomorphism.
Homotopy type of stabilizers of smooth functions with non-isolated singularities on surfaces
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abstract
The paper is devoted to the study of homotopy properties of stabilizers of smooth functions on oriented surfaces, i.e., groups of diffeomorphisms of surfaces preserving a given function. For some class of smooth functions which is a generalization of the class of Morse-Bott functions on oriented surfaces, the homotopy type of the connected component of the identity map of the stabilizer is completely described.
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Normal forms of functions with degenerate singularities on surfaces equipped with semi-free circle actions
A function with only Morse extrema and critical circles on a cylinder, torus, disk, or sphere always factors as a smooth value-reparametrization of the standard height, radius, or projection function, up to a diffeomorphism.