Morse functions, the Morse lemma, and the Morse index

Intuition Beginner

Imagine a smooth landscape — hills, valleys, and mountain passes — and a function that reads off the height at every point. The places where the ground is momentarily level, where you could set a marble down and it would not start to roll, are the special points. At a valley bottom every direction goes up. At a peak every direction goes down. At a pass some directions go up and some go down.

Morse theory is the study of how these level spots organise the whole shape. The remarkable fact is that a short list — how many valleys, how many passes, how many peaks — already constrains the global form of the landscape. You can recover deep features of a surface just by counting its level spots and recording, at each one, how many directions head downhill.

To make the counting reliable you want level spots that are "clean": no long flat ridges, no monkey saddles where three valleys meet. A Morse function is one whose level spots are all clean in this sense. Near a clean spot the landscape looks, after a gentle change of coordinates, like a sum of simple bowls and inverted bowls. The number of inverted bowls is the single number you record there.

Visual Beginner

Picture the height function on a vertical doughnut (a torus standing on its edge). Four spots are level: the bottom of the doughnut, the bottom of the inner hole, the top of the inner hole, and the very top.

At the bottom every direction climbs: a bowl, index $0$. At the two inner spots one direction climbs and one drops: saddles, index $1$. At the top every direction drops: an inverted bowl, index $2$. The four numbers $0,1,1,2$ are the fingerprint of the torus.

Worked example Beginner

Take the simplest curved surface, the round sphere, sitting in space, and let the function be the height above the floor. Only two spots are level: the south pole at the bottom and the north pole at the top. Everywhere else the surface tilts, so a marble would roll.

At the south pole, walk in any horizontal direction and you go up — both the east-west and the north-south directions climb. So the south pole is a bowl. The number of downhill directions is $0$.

At the north pole, walk in any direction and you descend — both directions drop. So the north pole is an inverted bowl. The number of downhill directions is $2$.

What this tells us: the height function on the sphere has exactly two clean level spots, with downhill-counts $0$ and $2$. Compare the torus, which had $0,1,1,2$. The extra pair of $1$'s is the signature of the hole. Counting level spots and their downhill numbers separates a sphere from a doughnut.

Check your understanding Beginner

Formal definition Intermediate+

Let $M$ be a smooth manifold of dimension $n$ and let $f\in C^{\infty }(M)$ be a smooth real-valued function. A point $p\in M$ is a critical point of $f$ if the differential $d{f}_p:T_{p}M\to \mathbb{R}$ vanishes; equivalently, in any chart $(U,x^1,\dots ,x^n)$ around $p$, all first partials $\partial f/\partial x^i$ vanish at $p$. A real number $c$ is a critical value if $f^{-1}(c)$ contains a critical point, and a regular value otherwise. The value $f(p)$ at a critical point is the critical level.

At a critical point the second-order data is intrinsic. Define the Hessian of $f$ at a critical point $p$ as the symmetric bilinear form

$$H_{p}f:T_{p}M\times T_{p}M\to \mathbb{R},H_{p}f(v,w)=v(\tilde{w}(f)),$$

where $\tilde{w}$ is any smooth vector field extending $w$ near $p$ and $v$ acts as a derivation on the function $\tilde{w}(f)$. This is well defined precisely because $d{f}_p=0$: for two extensions $\tilde{w},{\tilde{w}}^{\prime }$ of $w$ one has $v(\tilde{w}(f))-v({\tilde{w}}^{\prime }(f))=(\tilde{w}-{\tilde{w}}^{\prime })(v(f))$ at $p$ by symmetry of the Lie bracket pairing, and $v(f)=df(v)$ is the constant $0$ near $p$ along the critical condition, so the difference vanishes. The same computation gives symmetry $H_{p}f(v,w)=H_{p}f(w,v)$. In a chart, with $v={\sum }_i{v}^i{\partial }_i$ and $w={\sum }_j{w}^j{\partial }_j$,

$$H_{p}f(v,w)=\sum _{i,j}\frac{{\partial }^{2}f}{\partial x^i\partial x^j}(p)v^i{w}^j,$$

so the Hessian is represented by the ordinary symmetric matrix of second partials $({\partial }_i{\partial }_{j}f(p))$. Independence of the chart follows because the chart-change correction terms are multiples of the vanishing first partials.

A critical point $p$ is nondegenerate (a Morse critical point) if the Hessian $H_{p}f$ is a nondegenerate bilinear form, i.e. the matrix $({\partial }_i{\partial }_{j}f(p))$ is invertible. By Sylvester's law of inertia the Hessian then has a well-defined signature; the number of negative eigenvalues (counted with multiplicity) is the Morse index $\lambda (p)$, and the dimension of the kernel is the nullity. Nondegeneracy is exactly nullity $=0$. A function $f\in C^{\infty }(M)$ is a Morse function if every critical point of $f$ is nondegenerate.

Counterexamples to common slips

• The monkey saddle $f(x,y)=x^3-3x{y}^2$ has an isolated critical point at the origin with Hessian identically zero — degenerate, hence not Morse, even though the critical point is isolated. Isolatedness alone does not give nondegeneracy.
• The function $f(x,y)=x^2$ on ${\mathbb{R}}^2$ has the entire $y$-axis as critical points; the Hessian has nullity $1$ everywhere on it. Degenerate critical points may form positive-dimensional sets.
• The Hessian is intrinsic only at a critical point. At a non-critical point the matrix of second partials depends on the chart, so "the Hessian" is not defined there without extra structure (a connection).
• Index is the count of negative eigenvalues, not the determinant's sign. A $4\times 4$ Hessian of signature $(--++)$ and one of signature $(----)$ both have positive determinant but indices $2$ and $4$.

Key theorem with proof Intermediate+

The structural theorem of the subject is the local normal form. It follows the treatment in Milnor ^{[Milnor §2]} and Bott ^{[fasttrack-texts Critical points, the Hessian, the Morse lemma, and the index]}.

Theorem (Morse lemma). Let $p$ be a nondegenerate critical point of $f\in C^{\infty }(M)$, with Morse index $\lambda$. Then there is a chart $(U,y^1,\dots ,y^n)$ centred at $p$ (so $y^i(p)=0$) in which

$$f=f(p)-(y^1{)}^2-\cdots -(y^{\lambda }{)}^2+(y^{\lambda +1}{)}^2+\cdots +(y^n{)}^2$$

throughout $U$. In particular nondegenerate critical points are isolated, and the index $\lambda$ is a chart-independent invariant of the pair $(f,p)$.

Proof. Work in a chart centred at $p$ with coordinates $x=(x^1,\dots ,x^n)$ on a convex neighbourhood $V$ of $0$, and set $g(x)=f(x)-f(p)$, so $g(0)=0$ and $d{g}_0=0$.

Step 1: a smooth factorisation (Hadamard's lemma, second order). For $x\in V$, by the fundamental theorem of calculus along the segment $t\mapsto tx$,

$$g(x)={\int }_0^1\frac{d}{dt}g(tx)dt=\sum _i{x}^i{\int }_0^1{\partial }_{i}g(tx)dt=:\sum _i{x}^i{g}_i(x),$$

where each $g_i(x)={\int }_0^1{\partial }_{i}g(tx)dt$ is smooth and $g_i(0)={\partial }_{i}g(0)=0$. Applying the same device to each $g_i$,

$$g_i(x)=\sum _j{x}^j{\int }_0^1{\partial }_j{g}_i(sx)ds=:\sum _j{x}^j{h}_{ij}(x),$$

with $h_{ij}$ smooth. Hence

$$g(x)=\sum _{i,j}{x}^i{x}^j{h}_{ij}(x).$$

Replacing $h_{ij}$ by its symmetrisation $\frac{1}{2}(h_{ij}+h_{ji})$ leaves the sum unchanged, so assume $h_{ij}=h_{ji}$. Differentiating twice at $0$ gives ${\partial }_i{\partial }_{j}g(0)=2h_{ij}(0)$, so the symmetric matrix $H(x)=(h_{ij}(x))$ satisfies $H(0)=\frac{1}{2}({\partial }_i{\partial }_{j}g(0))$, which is invertible because $p$ is nondegenerate. By continuity $H(x)$ is invertible on a neighbourhood of $0$.

Step 2: diagonalise the family $H(x)$ by a smooth congruence (Gromoll splitting). Build new coordinates one axis at a time so that $g$ becomes $\pm (z^1{)}^2+\cdots$ with a remaining quadratic form in the later variables, the coefficients being smooth functions. Suppose inductively that on a neighbourhood of $0$ there are smooth coordinates $u=(u^1,\dots ,u^n)$, $u(0)=0$, in which

$$g=\pm (u^1{)}^2\pm \cdots \pm (u^{r-1}{)}^2+\sum _{i,j\ge r}{u}^i{u}^j{H}_{ij}(u),$$

with $H_{ij}=H_{ji}$ smooth and the matrix $(H_{ij}(0){)}_{i,j\ge r}$ invertible. After a constant linear change among $u^r,\dots ,u^n$ we may assume $H_{rr}(0)\ne 0$; by continuity $H_{rr}(u)\ne 0$ near $0$. Put $\epsilon =\mathrm{sign}{H}_{rr}(0)$ and define

$$v^r(u)=\sqrt{|H_{rr}(u)|}(u^r+\sum _{i>r}{u}^i\frac{H_{ir}(u)}{H_{rr}(u)}),$$

a smooth function of $u$ near $0$ since $H_{rr}$ is nonvanishing and of constant sign there. Then $\epsilon (v^r{)}^2$ collects exactly the terms of $g$ involving $u^r$: expanding,

$$\epsilon (v^r{)}^2=|H_{rr}|(u^r+{\sum }_{i>r}{u}^i\frac{H_{ir}}{H_{rr}}{)}^2\cdot \epsilon =H_{rr}(u^r{)}^2+2{\sum }_{i>r}{u}^r{u}^i{H}_{ir}+{\sum }_{i,j>r}{u}^i{u}^j\frac{H_{ir}{H}_{jr}}{H_{rr}},$$

so $g-\epsilon (v^r{)}^2={\sum }_{i,j>r}{u}^i{u}^j(H_{ij}-H_{ir}{H}_{jr}/H_{rr})$ involves only $u^{r+1},\dots ,u^n$ with smooth symmetric coefficients $H_{ij}^{\prime }(u)$. The map $u\mapsto (u^1,\dots ,u^{r-1},v^r,u^{r+1},\dots ,u^n)$ has Jacobian at $0$ equal to $\sqrt{|H_{rr}(0)|}\ne 0$ in the $r$-th diagonal slot and identity elsewhere, hence is a local diffeomorphism by the inverse function theorem. Renaming these as new coordinates advances the induction by one, and the Schur-complement matrix $(H_{ij}^{\prime }(0){)}_{i,j>r}$ is again invertible (it is the Schur complement of an invertible symmetric matrix). After $n$ steps,

$$g=\pm (z^1{)}^2\pm \cdots \pm (z^n{)}^2$$

in smooth coordinates $z$ centred at $0$.

Step 3: count the signs. Reorder the coordinates so the minus signs come first: $g=-(z^1{)}^2-\cdots -(z^k{)}^2+(z^{k+1}{)}^2+\cdots +(z^n{)}^2$. The Hessian of $g$ at $0$ in these coordinates is the diagonal matrix with $k$ entries $-2$ and $n-k$ entries $+2$, so its index is $k$. Because the index is the count of negative eigenvalues of the Hessian and the Hessian transforms by congruence $A\mapsto J^{\top }AJ$ under a coordinate change (with $J$ the Jacobian of the change at $0$), Sylvester's law of inertia forces $k=\lambda$, the index computed in the original chart. Setting $y^i=z^i$ and restoring $f=g+f(p)$ gives the stated normal form.

Isolatedness. In the normal-form chart $df=-2{\sum }_{i\le \lambda }{y}^{i}d{y}^i+2{\sum }_{i>\lambda }{y}^{i}d{y}^i$, which vanishes only at $y=0$. So $p$ is the only critical point in $U$: nondegenerate critical points are isolated, and on a compact manifold a Morse function has finitely many of them. $■$

Bridge. The normal form is the engine of every subsequent construction: it builds toward 03.02.31, where a nondegenerate critical point of index $\lambda$ is shown to attach a $\lambda$-cell as the sublevel set $\{f\le c\}$ crosses the critical level, turning a Morse function into a CW structure on $M$. The index, isolated and chart-independent here, appears again in 03.15.01 as the grading of the Morse chain complex, whose differential counts gradient trajectories between critical points of adjacent index. The Hadamard-factorisation technique reappears whenever a smooth function must be split off its vanishing jet — it is the local model behind the splitting lemma for degenerate critical points and behind the preparation theorems of singularity theory. And the genericity half of the theory, that Morse functions are dense, hands the deformation-invariance of the resulting Betti-number bounds to the Morse inequalities.

Exercises Intermediate+

Lean formalization Intermediate+

Mathlib has the ambient pieces — smooth manifolds, mfderiv, and the bilinear-form / quadratic-form inertia machinery — but no Morse layer, so this unit ships at lean_status: none. The statements below are the shape a Mathlib contribution would take; they do not yet compile against current Mathlib because the predicates IsCriticalPoint, hessian, and IsMorse are not defined.

-- Intended Mathlib-facing statements (not yet in Mathlib).

-- A point is critical when the manifold differential vanishes.
def IsCriticalPoint (f : M → ℝ) (p : M) : Prop := mfderiv I 𝓘(ℝ) f p = 0

-- The Hessian at a critical point as a symmetric bilinear form on T_p M.
-- Well-definedness uses IsCriticalPoint p (mfderiv f p = 0).
def hessian (f : M → ℝ) (p : M) (hp : IsCriticalPoint f p) :
    LinearMap.BilinForm ℝ (TangentSpace I p) := ...

-- A critical point is nondegenerate when its Hessian is nondegenerate.
def IsNondegenerate (f : M → ℝ) (p : M) (hp : IsCriticalPoint f p) : Prop :=
  (hessian f p hp).Nondegenerate

-- Morse lemma: a chart in which f has the standard ± quadratic normal form,
-- with k = morseIndex f p hp minus signs.
theorem morse_lemma
    (f : M → ℝ) (p : M) (hp : IsCriticalPoint f p)
    (hnd : IsNondegenerate f p hp) :
    ∃ (φ : PartialHomeomorph M (EuclideanSpace ℝ (Fin n))),
      p ∈ φ.source ∧
      ∀ x ∈ φ.source,
        f x = f p
          - ∑ i ∈ Finset.range (morseIndex f p hp), (φ x i)^2
          + ∑ i ∈ Finset.Ico (morseIndex f p hp) n, (φ x i)^2 := by
  sorry

The proof obligation behind morse_lemma is the parametrised splitting of Steps 1–2 above: a smooth symmetric matrix-valued function $H(x)$ with $H(0)$ invertible admits a smooth congruence to a constant signature matrix. That diagonalisation, together with a manifold Sard theorem for the density statement, is the concrete contribution target the Mathlib gap analysis names.

Advanced results Master

The Morse lemma is the nondegenerate case of a hierarchy. When the Hessian is degenerate, the Gromoll-Meyer splitting lemma factors $f$ near $p$ as a nondegenerate quadratic form on the range of the Hessian plus a function of the kernel variables alone: with $k$ the nullity, there are coordinates in which $f=f(p)-{\sum }_{i\le \lambda }{y}_i^2+{\sum }_{\lambda <i\le n-k}{y}_i^2+\varphi (y_{n-k+1},\dots ,y_n)$, where $\varphi$ has a critical point of nullity $k$ and vanishing $2$-jet at the origin. The Morse lemma is the case $k=0$, where the residual function $\varphi$ is absent. The splitting reduces the local study of any isolated critical point to the finite-dimensional "totally degenerate" piece, and underlies Thom's catastrophe classification: for $k=1$ and low codimension the residual $\varphi$ is, after a further change, one of the elementary catastrophe germs $y^3$ (fold), $y^4$ (cusp), and so on.

The Hessian carries more than its index. On a Riemannian manifold the gradient flow $\dot{x}=-\nabla f(x)$ has each nondegenerate critical point as a hyperbolic rest point whose linearisation is the Hessian operator; the stable manifold has dimension $n-\lambda$ and the unstable manifold dimension $\lambda$. When the descending and ascending manifolds of distinct critical points intersect transversally — the Morse-Smale condition, generic among metrics — their intersections are the moduli of gradient trajectories that build Morse homology, and the index becomes the homological degree. This is the bridge by which the purely local data of this unit acquires global homological force.

The infinite-dimensional extension is due to Palais and Smale ^{[Palais 1963 Topology 2]}. On a Hilbert manifold a $C^2$ function whose Hessian at each critical point is a self-adjoint Fredholm operator with finite index admits the Morse-Palais lemma: the same quadratic normal form holds in a Hilbert chart, now a sum and difference of squares over a Hilbert basis, with the index the dimension of the negative eigenspace. The decisive analytic input replacing compactness is the Palais-Smale condition (C): every sequence $(x_m)$ with $f(x_m)$ bounded and $\Vert df(x_m)\Vert \to 0$ has a convergent subsequence. The energy functional on the loop space — whose critical points are closed geodesics, and whose Hessian's index is computed by the Morse Index Theorem in terms of conjugate points — is the motivating example, and the reason Morse built the theory.

Genericity is sharpened by Sard's theorem. For a manifold $M\subset {\mathbb{R}}^N$ the family of height functions $f_a(x)=⟨a,x⟩$, $a\in S^{N-1}$, is Morse for almost every direction $a$: the degenerate directions are the critical values of the Gauss-type map sending a point of $M$ to its normal directions, a measure-zero set by Sard. More generally, within $C^{\infty }(M)$ with the Whitney topology the Morse functions are open and dense, so any smooth function is a small perturbation of a Morse function and any two Morse functions are joined by a path that is Morse outside finitely many birth-death moments. This density is what makes the critical-point counts deformation-stable, and hence genuine invariants.

Synthesis. The index organises a web of identifications. It builds toward the handle-attachment theorem of 03.02.31 and the Morse-complex grading of 03.15.01, where it becomes the homological degree; it specialises, through the Hessian-as-Fredholm-operator viewpoint, to the conjugate-point count of the Morse Index Theorem for geodesics; it reappears in 03.12.10 as the dimension of the cell a critical point contributes to the CW structure of $M$; through the gradient flow it is the dimension of the unstable manifold, tying the local linear-algebra invariant to the global dynamics; and through Sard-theoretic genericity it is stable under perturbation, which is the precondition for the Morse inequalities $b_k(M)\le c_k$ relating Betti numbers to critical-point counts. The same Hadamard-Gromoll splitting that proves the lemma is, in its degenerate form, the local backbone of singularity and catastrophe theory.

Full proof set Master

The Morse lemma is proved in full in the Key theorem section (Steps 1–3, with isolatedness). Here are the remaining Master-level claims.

Chart-independence and the inertia computation. Under a coordinate change $y=\psi (x)$ with Jacobian $J=D\psi (0)$, the second-partials matrix transforms at a critical point as $A\mapsto J^{-\top }A{J}^{-1}$ plus a term ${\sum }_k{\partial }_{k}f\cdot {\partial }^2{\psi }^k$ that vanishes because ${\partial }_{k}f(0)=0$. Congruence preserves the count of negative eigenvalues by Sylvester's law of inertia, so the index $\lambda$ is chart-independent; the same identity shows the Hessian bilinear form on $T_{p}M$ is intrinsic, recovering the coordinate-free definition. $■$

Isolatedness on compact $M$ gives finiteness. By the Morse lemma each nondegenerate critical point has a neighbourhood containing no other critical point. The critical set is closed (it is $\{df=0\}$). A closed, discrete subset of a compact space is finite. Hence a Morse function on a compact manifold has finitely many critical points. $■$

Gromoll-Meyer splitting (statement; proof sketch). Write the Hessian $A=H_{p}f$ with kernel $K$ of dimension $k$ and orthogonal complement $K^{\perp }$. The Hadamard factorisation gives $f-f(p)=⟨H(x)x,x⟩$ with $H(0)=\frac{1}{2}A$. Apply Steps 1–2 of the Morse-lemma diagonalisation only to the $K^{\perp }$ block, where $H(x)$ restricted to $K^{\perp }$ stays invertible near $0$; this splits off $-{\sum }_{i\le \lambda }{y}_i^2+{\sum }_{\lambda <i\le n-k}{y}_i^2$. The implicit function theorem applied to $\partial f/\partial y_{K^{\perp }}=0$ solves the $K^{\perp }$-variables as smooth functions of the $K$-variables, and substituting yields the residual $\varphi$ on $K$ with vanishing $2$-jet. The full proof is in Gromoll-Meyer 1969 ^{[Milnor §2 (splitting lemma context)]}. $■$

Almost-every height function is Morse (Sard). For $M\subset {\mathbb{R}}^N$ embedded, define $\nu :NM\to {\mathbb{R}}^N$ on the normal bundle by $(x,v)\mapsto x+v$ (the endpoint map). A direction $a$ fails to be Morse for $f_a(x)=⟨a,x⟩$ exactly when $a$ is a focal/critical value of a map built from $\nu$; by Sard's theorem the critical values form a measure-zero set, so almost every $a$ gives a Morse function. The density of Morse functions in the Whitney $C^{\infty }$ topology follows by the Thom transversality theorem applied to the $1$-jet section $j^{1}f$ meeting the zero section transversally, transversality to the diagonal of the $2$-jet giving nondegeneracy. $■$

Connections Master

The handle/cell-attachment theorem 03.02.31 is the direct successor: passing a nondegenerate critical level of index $\lambda$ changes the homotopy type of the sublevel set $\{f\le c\}$ by attaching a single $\lambda$-cell, with the Morse lemma's normal form supplying the explicit attaching map. Everything proved here about the index as a chart-independent invariant is exactly the data that theorem consumes; without isolatedness and the well-defined index, the cell count would not be defined.

Morse homology 03.15.01 promotes the index to a homological grading. The Morse chain group in degree $k$ is the free abelian group on the index-$k$ critical points of a Morse-Smale pair $(f,g)$, and the differential counts gradient trajectories from index-$k$ to index-$(k-1)$ critical points. The fact established here — that nondegenerate critical points are isolated and finite on compact $M$ — is what makes these chain groups finitely generated, and the Hessian's hyperbolicity is what makes the trajectory moduli finite-dimensional.

The CW complex 03.12.10 is the topological output: a Morse function endows $M$ with the homotopy type of a CW complex having one $\lambda$-cell per index-$\lambda$ critical point. The link is dimensional — the index computed here is precisely the dimension of the attached cell — so the Morse-theoretic Betti-number bounds become CW-cellular-chain computations, and the Euler-characteristic identity $\chi (M)={\sum }_p(-1{)}^{\lambda (p)}$ is the Morse incarnation of the cellular Euler characteristic.

Historical & philosophical context Master

Marston Morse introduced nondegenerate critical points and the index in his 1925 paper Relations between the critical points of a real function of $n$ independent variables (Transactions of the American Mathematical Society 27, 345–396), where the inequalities bounding Betti numbers by critical-point counts first appear. He consolidated the theory, including the variational calculus "in the large" for geodesics, in The Calculus of Variations in the Large (American Mathematical Society Colloquium Publications 18, 1934). The local normal form now called the Morse lemma is in that lineage, with the clean modern proof via the parametrised Hadamard factorisation due to the exposition tradition that runs through Bott's lectures and Milnor's book.

John Milnor's Morse Theory (Annals of Mathematics Studies 51, 1963), based on lectures with notes by Spivak and Wells, gave the subject its standard form: critical points and the index in Part I §2, the handle-attachment and homotopy results, and the application to the Bott periodicity theorem in Part IV. Richard Palais extended the theory to infinite-dimensional Hilbert manifolds in Morse theory on Hilbert manifolds (Topology 2, 1963, 299–340), proving the Morse-Palais lemma; together with Stephen Smale's condition (C), this made the loop-space energy functional a legitimate Morse function and recovered Morse's geodesic results inside the new framework. Raoul Bott's 1982 survey lectures place the local theory of this unit within the broader arc from periodicity to Yang-Mills.

The degenerate theory — Gromoll and Meyer's splitting lemma (1969) and René Thom's catastrophe classification (Stabilité structurelle et morphogénèse, 1972) — extends the normal-form program past nondegeneracy, classifying the simplest degenerate germs. Sard's theorem (Arthur Sard, 1942) supplies the genericity that makes Morse functions dense and the resulting invariants deformation-stable.

Bibliography Master

@article{Morse1925,
  author  = {Morse, Marston},
  title   = {Relations between the critical points of a real function of $n$ independent variables},
  journal = {Transactions of the American Mathematical Society},
  volume  = {27},
  number  = {3},
  pages   = {345--396},
  year    = {1925}
}

@book{Morse1934,
  author    = {Morse, Marston},
  title     = {The Calculus of Variations in the Large},
  series    = {American Mathematical Society Colloquium Publications},
  volume    = {18},
  publisher = {American Mathematical Society},
  year      = {1934}
}

@book{Milnor1963,
  author    = {Milnor, John},
  title     = {Morse Theory},
  series    = {Annals of Mathematics Studies},
  volume    = {51},
  publisher = {Princeton University Press},
  year      = {1963},
  note      = {Based on lecture notes by M. Spivak and R. Wells}
}

@article{Palais1963,
  author  = {Palais, Richard S.},
  title   = {Morse theory on {H}ilbert manifolds},
  journal = {Topology},
  volume  = {2},
  pages   = {299--340},
  year    = {1963}
}

@article{GromollMeyer1969,
  author  = {Gromoll, Detlef and Meyer, Wolfgang},
  title   = {On differentiable functions with isolated critical points},
  journal = {Topology},
  volume  = {8},
  pages   = {361--369},
  year    = {1969}
}

@incollection{Bott1982,
  author    = {Bott, Raoul},
  title     = {Lectures on {M}orse theory, old and new},
  booktitle = {Bulletin of the American Mathematical Society (N.S.)},
  volume    = {7},
  number    = {2},
  pages     = {331--358},
  year      = {1982}
}

@article{Sard1942,
  author  = {Sard, Arthur},
  title   = {The measure of the critical values of differentiable maps},
  journal = {Bulletin of the American Mathematical Society},
  volume  = {48},
  pages   = {883--890},
  year    = {1942}
}