Geodesic
On Functions Whose All Critical Points Are Contained in a Ball
Funkcionalʹnyj analiz i ego priloženiâ, Tome 36 (2002) no. 4, pp. 80-83.

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In the present note, we answer the following question posed by Arnold. Consider a function with finitely many critical points on a compact connected manifold without boundary. Suppose that a ball embedded in the manifold contains all critical points of the function. Is it possible to reconstruct the manifold by a restriction of the function to the ball? It turns out that one can reconstruct only the Euler characteristic of the manifold.
Keywords: Morse function, gradient-like vector field. 
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P. E. Pushkar'. On Functions Whose All Critical Points Are Contained in a Ball. Funkcionalʹnyj analiz i ego priloženiâ, Tome 36 (2002) no. 4, pp. 80-83. http://geodesic.mathdoc.fr/item/FAA_2002_36_4_a9/

[1] Arnold V. I., Zadachi Arnolda, FAZIS, M., 2000 | MR

[2] Milnor Dzh., Teorema ob $h$-kobordizme, Mir, M., 1969 | MR