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Eighth problem of the Arnold trivium

Timur Kenzhaev

Abstract

We give a full and detailed solution of eighth Arnold's trivium problem. We find critical points of a smooth function on a given one-parametric two-dimensional surface with Lagrange multipliers method. Basic Morse theory and Poincare-Hopf theorem makes it possible to determine a genus of this surface and gives a beautiful training example of Morse surgery on a two-dimensional surface of genus $g = 6$.

Eighth problem of the Arnold trivium

Abstract

We give a full and detailed solution of eighth Arnold's trivium problem. We find critical points of a smooth function on a given one-parametric two-dimensional surface with Lagrange multipliers method. Basic Morse theory and Poincare-Hopf theorem makes it possible to determine a genus of this surface and gives a beautiful training example of Morse surgery on a two-dimensional surface of genus .
Paper Structure (9 sections, 2 theorems, 15 equations)

This paper contains 9 sections, 2 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Parameter $C$ variation
  3. Surface smoothness
  4. Critical points
  5. Points of the first type
  6. Points of the second type
  7. Non-smooth case
  8. Results and consequences
  9. Genus of the surface

Key Result

Theorem 4.1

Morse_Theory Let $f$ be a smooth real-valued function on a manifold $M$. Let $a < b$ and suppose that the set $f^{-1}\left([a, b]\right)$, consisting of all $p\in M$ with $a\leq f(p)\leq b$, is compact, and contains no critical points of $f$. Then $M^{a}$ is diffeomorphic to $M^b$. Furthermore, $M^a

Theorems & Definitions (2)

  • Theorem 4.1
  • Theorem 4.2