MathJax(?): Can't find handler for document MathJax(?): Can't find handler for document ( &#x2202; 2 f ( p ) &#x2202; x i &#x2202; x j ) 1 &#x2264; i , j &#x2264; nHow can we deduce that f only has finitely many non-degenerate critical points by this function only has non-degenerate critical point? And how can we use the compact manifold’s properties to solve the question?

Question

MathJax(?): Can&#039;t find handler for document MathJax(?): Can&#039;t find handler for document                    (                                                                      &amp;#x2202;                  2                                f                (                p                )                                            &amp;#x2202;                                  x                  i                                &amp;#x2202;                                  x                  j                                                              )                  1  &amp;#x2264;  i  ,  j  &amp;#x2264;  nHow can we deduce that f only has finitely many non-degenerate critical points by this function only has non-degenerate critical point? And how can we use the compact manifold’s properties to solve the question?

Tasinazzokbc · Accepted Answer

H  e  s  s  (  x  )) (the matrix in the question is the second differential). Let   U be a chart containing x that we identify to an open subset of             R        n   the restriction of   d  f to   U is a map   U  →            R        n   and   H  e  s  s  (  x  ) is the differential of this map. The fact that   H  e  s  s  (  x  ) is invertible implies that   d  f is locally invertible, therefore there exists   d  f open such that the restriction   d      f          ∣              U        x             is injective.Suppose that there exists an infinite numbers of critical points       x    1    ,  .  .  .  ,      x    n    ,  .  .  . this sequence has an accumulation point   y.   y is also critical since   d  f is continuous, but there does not exist a neighborhood   V of   V such that the restriction of   d  f to   V is invertible. Contradiction.

(delta^2 f(p)delta x_i delta x_j) 1<=i,j<=n How can we deduce that f only has finitely many non-degenerate critical points by this function only has non-degenerate critical point? And how can we use the compact manifold’s properties to solve the question?

Answered question

Answer & Explanation

New Questions in Differential Calculus