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  "body": "\n## Definition\n\nLet $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$, and let $f \\colon M \\to \\R$ be a [[smooth function|0V33BNKZ9S]].\nA point $p \\in M$ is a [[critical point|0YTCMFB4HC]] of $f$ if $df_p = 0$.\nA critical point $p$ is *nondegenerate* if, in some chart $(x_1, \\dots, x_n)$ around $p$, the [[Hessian matrix|0693Y2METT]] $\\big(\\partial^2 f / \\partial x_i \\partial x_j(p)\\big)$ is invertible.\nThe function $f$ is a *Morse function* if every critical point of $f$ is nondegenerate.\nThe *index* of a nondegenerate critical point is the number of negative eigenvalues of its Hessian.\n\n## Remarks\n\nAt a critical point the Hessian transforms under a change of chart by $H \\mapsto A^T H A$ with $A$ invertible, so nondegeneracy and the index do not depend on the chart.\nExample: $f \\colon \\R^2 \\to \\R$, $f(x, y) = x^2 - y^2$, is a Morse function.\nIts only critical point is the origin, where the Hessian is $\\begin{pmatrix} 2 \u0026 0 \\\\ 0 \u0026 -2 \\end{pmatrix}$, which is invertible; the index is $1$.\nNon-example: $g \\colon \\R \\to \\R$, $g(x) = x^3$, is not a Morse function.\nIts critical point $0$ has Hessian $g''(0) = 0$, which is not invertible.\nNondegenerate critical points are isolated, so on a compact manifold a Morse function has only finitely many critical points.\nOn a compact manifold the Morse functions form an open dense subset of $C^\\infty(M, \\R)$.\n",
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