Quadratic Morse-Bott Function

• Quadratic Morse-Bott function is a smooth function on a manifold whose critical locus forms nondegenerate submanifolds with locally quadratic behavior in the normal directions.
• It is applied in differential topology, symplectic geometry, and representation theory to stratify spaces and compute Morse indices via precise Hessian analysis.
• Its gradient flows and optimal Łojasiewicz inequalities enable exponential convergence and deformation retracts, facilitating topological decompositions and cohomological insights.

A quadratic Morse-Bott function is a smooth function on a manifold whose critical locus is a union of nondegenerate critical submanifolds, and in local coordinates transverse to these submanifolds, the function is quadratic. The precise Morse-Bott condition requires that at each critical point, the kernel of the Hessian matches the tangent space to the critical submanifold, and the Hessian is nondegenerate on the normal bundle. Quadratic Morse-Bott functions constitute a foundational class of functions in differential topology, symplectic geometry, and representation theory, providing canonical local models for singularities with non-isolated critical points and enabling fine stratifications of spaces.

1. Local Structure and Morse-Bott Condition for Quadratic Forms

Let $M$ be a finite- or infinite-dimensional manifold (Banach or Hilbert, as appropriate), $f : M \to \mathbb{R}$ a smooth (often analytic) function, and let $\operatorname{Crit}(f) \subset M$ denote the critical locus. The Morse-Bott property at $x_0 \in \operatorname{Crit}(f)$ is defined by the condition that $\operatorname{Crit}(f)$ is a submanifold near $x_0$, the Hessian $D^2f(x_0)$ has kernel equal to $T_{x_0}\operatorname{Crit}(f)$, and the Hessian is nondegenerate on a complement.

For a quadratic Morse-Bott function, in local coordinates $(z,w) \in B_0 \oplus W$ near a critical point, one has

$$f(\Phi(z,w)) = f(x_0) + \frac{1}{2} \langle Az, z\rangle + R(w)$$

where $A$ is invertible, $R(w)$ vanishes to order $>2$ at $w=0$. This canonical splitting (Morse-Bott Lemma (Feehan, 2018)) demonstrates quadratic behavior transverse to the critical manifold. The sharp Łojasiewicz gradient inequality $\|\nabla f(x)\| \geq C|f(x) - f(x_0)|^{1/2}$ holds, with exponent $1/2$ being optimal for the quadratic model.

Finite-dimensional examples include $f(x, y) = x^2$ (critical locus a line, quadratic in normal), $f(x, y, z) = x^2+y^2$ (critical locus a line, quadratic in the plane), and $f(x, y) = x^2-y^2$ (standard Morse function at an isolated critical point) (Feehan, 2018). In analytic Banach space settings, these local models and inequalities fully characterize the Morse-Bott property.

2. Prototypical Geometric Realizations

2.1 Squared Distance Functions

Let $(M, g)$ be a complete Riemannian manifold, $N \subset M$ a closed embedded submanifold. The squared distance function $f(x) = d(x, N)^2$ is a quadratic Morse-Bott function on $M \setminus \operatorname{Cu}(N)$, where $\operatorname{Cu}(N)$ is the cut locus of $N$. The critical locus is $N$ itself, and at each $p \in N$, the Hessian is nondegenerate in the normal bundle $\nu_pN$ (eigenvalues +2), vanishing in $T_pN$ (Basu et al., 2020).

In tubular (Fermi) coordinates $(x, y) \in N \times \mathbb{R}^{n-k}$, $f(x, y) = \|y\|^2 + O(\|y\|^3)$. The gradient flow is $\dot{y} = -2y + O(\|y\|^2)$, asymptotically retracting a neighborhood onto $N$. The function is smooth and Morse-Bott away from the cut locus; the Hessian becomes singular on $\operatorname{Cu}(N)$. For $N = S^n \subset \mathbb{R}^{n+1}$, $f(p) = (\|p\| - 1)^2$, yielding radial contraction (Basu et al., 2020).

2.2 Quadratic Trace Functions on Lie Groups and Homogeneous Spaces

On $O(n)$, the quadratic trace function $f(X) = \operatorname{Tr}(A X B X^T)$, with $A,B$ real diagonal matrices, is Morse-Bott. The critical locus consists of submanifolds classified via “perfect fillings” (block diagonalizations corresponding to eigenvalue multiplicities), and the Morse-Bott index is given by a combinatorial sum over rectangles in the block table (Bozma et al., 2018).

On quaternionic Stiefel manifolds $$X_{n,k} = \{x \in \mathbb{H}^{n \times k}: x^* x = I_k\} \cong Sp(n)/Sp(n-k)$$, the function $f(x) = \operatorname{Tr}(P^* P)$, with $$x = \begin{pmatrix} T \ P \end{pmatrix}$$, is quadratic Morse-Bott. Critical submanifolds $E_q$, indexed by the rank of $P^*P$, are fiber bundles over products of quaternionic Grassmannians, and the Morse index is $4(n - 2k + q)q$ (Macías-Virgós et al., 2020).

On real Grassmannians $Gr(k;V)$ of a symplectic vector space $(V, \omega)$ with a compatible complex structure $J$, the quadratic Morse-Bott function $f(W) = \frac{1}{2} \|[P_W, J]\|^2$ classifies $W$ according to the dimension triple $(n_0, n_+, n_-)$ from the canonical orthogonal splitting, with critical loci forming homogeneous spaces for $U(n)$ (Kim, 23 Jan 2026). Negative gradient flow coincides with $Sp(V)$-orbits, with stable manifolds retracting onto these critical loci.

3. Critical Set Structure and Hessian Nondegeneracy

For quadratic Morse-Bott functions, the critical set is a closed (analytic) submanifold near each point, and the Hessian is transverse nondegenerate. Explicitly, if $f: M \to \mathbb{R}$ is quadratic Morse-Bott near $x_0$, then

• $\ker D^2f(x_0) = T_{x_0}\operatorname{Crit}(f)$
• The range of $D^2f(x_0)$ is a closed complement, and $D^2f(x_0)$ is invertible on this complement

The explicit computation of the Hessian in local normal coordinates confirms that all nontrivial directions yield eigenvalues of $\pm 2$ (or appropriate multiples), and the Morse index at each component may be computed algebraically or combinatorially, as in the perfect fillings approach for orthogonal groups (Bozma et al., 2018) or via block decompositions in Stiefel/Grassmannian cases (Macías-Virgós et al., 2020, Kim, 23 Jan 2026).

4. Gradient Flows and Deformation Retractions

Quadratic Morse-Bott functions admit negative gradient flows that are explicitly computable in normal coordinates. For the squared distance function, the gradient flow follows geodesics in the normal bundle, converging exponentially to the submanifold. In the context of homogeneous spaces (e.g., Stiefel manifolds, Grassmannians), the flows are equivariant under group actions and induce strong deformation retracts of open subsets (e.g., $M \setminus \operatorname{Cu}(N)$, open $Sp(V)$-orbits) onto critical submanifolds (Basu et al., 2020, Macías-Virgós et al., 2020, Kim, 23 Jan 2026).

These flows enable Morse-Bott stratifications of the ambient space, with each stratum being the stable manifold of a critical component. The flows respect the symmetries of the space, preserving group orbits and allowing for combinatorial and topological interpretations of the stratification.

5. Topological and Spectral Consequences

The Morse-Bott structure of quadratic functions provides explicit decompositions of the (co)homology of the ambient space via spectral sequences or direct summations indexed by the critical manifolds. For instance, cohomology of $O(n)$ and $SO(n)$ may be computed via Morse-Bott decompositions associated to critical loci of quadratic trace functions, with the Morse index providing the grading shift and the critical component determining the summand (Bozma et al., 2018).

For spaces such as Grassmannians of symplectic vector spaces, the Morse-Bott critical loci correspond to classical flag varieties (e.g., Lagrangian, isotropic, or coisotropic cases), and the stable manifold/cellular decomposition recovers the homotopy type and the cell attachments of these spaces (Kim, 23 Jan 2026). Similar decompositions hold for Stiefel manifolds via fibrations over products of Grassmannians (Macías-Virgós et al., 2020). In analytic infinite-dimensional contexts, the Morse-Bott property implies sharp gradient inequalities of Łojasiewicz type, controlling convergence rates for gradient flows and singularity analysis (Feehan, 2018).

6. Analytic Foundations and Łojasiewicz Inequalities

Quadratic Morse-Bott singularities are characterized analytically by the gradient inequality $\|\nabla f(x)\| \geq C |f(x) - f(x_0)|^{1/2}$ in a neighborhood of $x_0$; this exponent $1/2$ is sharp and, conversely, the holding of this inequality for an analytic function implies the Morse-Bott property at $x_0$ (Feehan, 2018). The Morse-Bott Lemma provides diffeomorphic coordinates linearizing the critical set and quadraticizing the function transverse to it.

In Banach and Hilbert space settings, this analytic structure supports applications to nonlinear PDEs (harmonic map energy, Yang–Mills, mean curvature flow), where the Morse-Bott property determines the local landscape of functionals near nonisolated critical submanifolds.

7. Connections, Generalizations, and Examples

Quadratic Morse-Bott functions arise naturally in a range of contexts:

• Distance-squared functions to submanifolds in Riemannian geometry (with singularities at the cut locus) (Basu et al., 2020)
• Invariant quadratic trace-type functions on algebraic and Lie groups, stratifying group manifolds and enabling cohomological calculations (Bozma et al., 2018)
• Quadratic energy-type functions on Grassmannians and Stiefel manifolds, providing global stratifications and explicit gradient flows (Kim, 23 Jan 2026, Macías-Virgós et al., 2020)
• Analytical characterizations in infinite dimensions, crucial for gradient flow and singularity analysis (Feehan, 2018)

These models serve as the normal forms for functions with nonisolated, nondegenerate critical manifolds, supporting a unified approach to Morse-Bott theory in both finite and infinite dimensions. The explicit local and global structures, gradient flows, and topological decompositions provided by quadratic Morse-Bott functions are central to singularity theory, global analysis, and the topology of homogeneous spaces.