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  "body": "\n## Definition\n\nLet $M$ be a [[smooth manifold|F45VQ47GPV]] and let $f \\colon M \\to \\R$ be a [[smooth function|0V33BNKZ9S]] with set of [[critical points|0YTCMFB4HC]] $\\operatorname{Crit}(f) = \\set{p \\in M : df_p = 0}$.\nThe function $f$ is a *Morse–Bott function* if:\n\n1. each connected component of $\\operatorname{Crit}(f)$ is a closed submanifold of $M$; and\n2. for each $p \\in \\operatorname{Crit}(f)$, the kernel of the Hessian $\\operatorname{Hess}_p f$ on $T_p M$ equals $T_p C$, where $C$ is the component of $\\operatorname{Crit}(f)$ through $p$.\n\n## Remarks\n\nThe Hessian is defined at critical points as in the definition of a [[nondegenerate critical point|MGPPCDFQDG]].\nSince $f$ is constant on each component $C$, the Hessian always vanishes on $T_p C$; the second condition says it is nondegenerate on the normal space $T_p M / T_p C$.\nThe *index* of a critical component is the number of negative directions of the Hessian on this normal space.\nA [[Morse function|KPQ96640ZJ]] is a Morse–Bott function whose critical components are points.\n\nExample: $f(x, y) = x^2$ on $\\R^2$ has $\\operatorname{Crit}(f) = \\set{x = 0}$, a line.\nAt $(0, y)$ the Hessian matrix is $\\operatorname{diag}(2, 0)$, whose kernel is spanned by $\\partial_y$, the tangent direction of the line; so $f$ is Morse–Bott, with critical line of index $0$.\n\nNon-example: $g(x, y) = x^4$ also has $\\operatorname{Crit}(g) = \\set{x = 0}$, a submanifold, but its Hessian at $(0, y)$ is $0$.\nIts kernel is all of $\\R^2$, not the tangent line, so the second condition fails.\nAnother non-example is $h(x, y) = x^2 y^2$, whose critical set is the union of the two axes, which is not a submanifold near the origin.\n",
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