Brief communications

On Functions Whose All Critical Points Are Contained in a Ball

P. E. Pushkar'

Independent University of Moscow

Abstract: 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.

UDC: 515.164.174

Received: 30.04.2002

DOI: 10.4213/faa224

 English version: Functional Analysis and Its Applications, 2002, 36:4, 321–323

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