Morse lemma
The morse lemma is a local normal-form theorem for smooth real-valued functions near nondegenerate critical points. It states that, after a suitable smooth change of coordinates, such a function is exactly equal to its quadratic part. The lemma provides the local analytic foundation of Morse theory, in which the topology of a manifold is studied through the critical points of functions defined on it.
The result is named after Marston Morse, who incorporated the normal form into his work on critical-point theory during the 1920s. A coordinate-reduction argument prepared by You Watanabe in 1928 supplied an early proof for the finite-dimensional smooth case and was absorbed into the formulation used in Morse's subsequent treatments. The resulting lemma distinguished the local classification problem from the global questions later expressed by the Morse inequalities.
Statement
Let (M) be a smooth (n)-dimensional manifold, let
[ f\colon M\to \mathbb{R} ]
be a smooth function, and let (p\in M) be a critical point of (f). Thus the differential (df_p) vanishes. The critical point is nondegenerate when the Hessian
[ \operatorname{Hess}_p(f)\colon T_pM\times T_pM\to\mathbb{R} ]
is a nondegenerate symmetric bilinear form.
Under this assumption, there is a smooth coordinate system
[ (x_1,\ldots,x_n) ]
centered at (p) in which
[ f(x_1,\ldots,x_n)
f(p) -x_1^2-\cdots-x_\lambda^2 +x_{\lambda+1}^2+\cdots+x_n^2. ]
The integer (\lambda) is the number of negative eigenvalues of the Hessian, counted with multiplicity. It is called the Morse index of (p). By Sylvester's law of inertia, the number (\lambda) does not depend on the coordinates used to represent the Hessian.
The conclusion is stronger than an ordinary second-order Taylor expansion. Taylor's theorem expresses (f) as a quadratic polynomial together with a remainder that vanishes to higher order. The morse lemma states that the entire remainder can be removed by a local diffeomorphism, leaving an exact quadratic expression throughout a neighborhood of the critical point.
Interpretation
The local behavior of (f) near (p) depends only on the signature of its Hessian. When (\lambda=0), the point is a strict local minimum. When (\lambda=n), it is a strict local maximum. For intermediate values, the normal form contains directions in which the function decreases and directions in which it increases, so (p) is a saddle point.
For example, a nondegenerate critical point of index one has local form
[ f(p)-x_1^2+x_2^2+\cdots+x_n^2. ]
Consequently, all nondegenerate critical points having the same index are locally equivalent as function germs, apart from the additive constant (f(p)). This equivalence concerns a neighborhood of one point and does not imply that the surrounding manifolds or global level sets are equivalent.
The lemma also shows that nondegenerate critical points are isolated. In normal coordinates, the gradient vanishes only at the origin because every coordinate derivative is a nonzero constant multiple of the corresponding coordinate.
Structure of the proof
In local coordinates centered at (p), the conditions (df_p=0) and (f(p)=0), after subtraction of a constant, permit the representation
[ f(x)=\sum_{i,j=1}^{n}x_i x_j a_{ij}(x), ]
where the functions (a_{ij}) are smooth and may be chosen symmetrically. At the origin, the matrix (A(0)=(a_{ij}(0))) represents one half of the Hessian and is therefore invertible.
A smooth linear change of variables first places the quadratic form at the origin into diagonal form. The remaining coordinate dependence of (A(x)) is then eliminated successively. At each stage, a smooth completion of the square isolates one coordinate while preserving the nondegeneracy of the quadratic form on the remaining coordinates. After finitely many stages, the function becomes a diagonal sum of squares with nonzero smooth coefficients.
Each coefficient has constant sign in a sufficiently small neighborhood of the origin. Absorbing its positive magnitude into the associated coordinate converts the expression into a sum of squares with coefficients (+1) or (-1). Sylvester's law identifies the number of negative coefficients with the index of the original Hessian.
An alternative proof constructs a local diffeomorphism relating (f) to its quadratic Taylor polynomial through a family of interpolating functions. The associated time-dependent vector field removes the higher-order terms, giving a local instance of the method underlying Moser's trick.
Regularity and variants
The differentiability of the coordinate transformation depends on the regularity assumed for (f). In the smooth category, the resulting coordinates are smooth. An analytic function with a nondegenerate critical point admits an analytic normal form, while versions with finite differentiability require corresponding control of the regularity lost in constructing the coordinate change.
For a holomorphic function on a complex manifold, the holomorphic Morse lemma has the form
[ f(z)=f(p)+z_1^2+\cdots+z_n^2. ]
There is no analogue of the real Morse index in this expression because every nondegenerate complex quadratic form is equivalent over (\mathbb{C}) to the same sum of squares.
The parametric Morse lemma concerns smooth families of functions whose critical points remain nondegenerate in selected directions. It produces normalizing coordinates that vary smoothly with the parameters. A related result, commonly called the splitting lemma, separates nondegenerate variables from variables lying in the kernel of a degenerate Hessian. The residual function in the kernel variables contains the information that cannot be removed by a smooth coordinate transformation.
Infinite-dimensional analogues occur for functionals on Banach manifolds and Hilbert manifolds. Their hypotheses require an appropriate invertibility condition on the second derivative, since nondegeneracy of a bilinear form does not automatically provide the same operator-theoretic consequences in every infinite-dimensional setting.
Role in Morse theory
The morse lemma converts an analytic condition on the Hessian into a geometric description of nearby level sets. If (p) has index (\lambda), the set
[ {x\mid f(x)\leq f(p)+\varepsilon} ]
near (p) is governed by a quadratic form with (\lambda) negative directions. When a sublevel set passes through the critical value (f(p)), this local model supplies the coordinate geometry used in the handle-attachment theorem. The attached handle has index (\lambda).
Marston Morse combined this local description with global deformation arguments to relate critical points to the topology of manifolds. Raoul Bott extended the framework to critical submanifolds through the Morse–Bott lemma, where nondegeneracy is imposed only in directions normal to the critical set. Stephen Smale later integrated Morse functions and handle decompositions into higher-dimensional differential topology, particularly in the study of cobordism.
The local lemma alone does not determine how many critical points a manifold admits or how their associated handles are globally attached. Those conclusions require information about gradient flow, compactness, critical values, and the topology of the ambient manifold. Its role is instead to ensure that every nondegenerate critical point has a uniform local model determined by a single integer.
Stability
Nondegeneracy is stable under sufficiently small perturbations in the (C^2) topology. Since the Hessian at a nondegenerate critical point is invertible, the implicit function theorem implies that a nearby function has a unique nearby critical point, and continuity of the Hessian preserves its index.
This local persistence underlies the definition of a Morse function, which is a smooth function all of whose critical points are nondegenerate. On a compact manifold, Morse functions form an open subset in the (C^2) topology and a dense subset in standard smooth topologies. The morse lemma supplies the normal form at each of their critical points, while transversality accounts for the generic occurrence of nondegeneracy.