Obata-type Rigidity Theorem

• Obata-type Rigidity Theorem is a fundamental result in geometric analysis that characterizes global symmetry properties under precise spectral or PDE constraints.
• It extends classical rigidity from Riemannian manifolds to sub-Riemannian, CR, Kähler, and metric measure spaces by exploiting Lichnerowicz-type bounds and overdetermined Hessian equations.
• The theorem employs techniques such as Bochner identities, eigenfunction characterization, and warped-product models to achieve classification of geometric structures.

The Obata-type rigidity theorem is a central result in geometric analysis characterizing global symmetry properties of geometric spaces under sharp spectral or PDE constraints. Originally formulated for Riemannian manifolds, the theorem and its analogues have been widely extended to sub-Riemannian, Kähler, CR (Cauchy–Riemann), stratified, and metric measure spaces, as well as to manifolds with boundary, yielding powerful classification statements for spaces attaining the equality case in Lichnerowicz-type lower bounds or solving specific overdetermined Hessian equations.

1. Classical Obata Rigidity in Riemannian Geometry

The classical Obata rigidity theorem asserts: Let $(M^n, g)$ be a complete Riemannian manifold. If there exists a nonconstant smooth function $u$ such that

$\nabla^2 u + \lambda u\, g = 0$

for some constant $\lambda>0$, then $(M,g)$ is isometric to the round sphere of radius $1/\sqrt{\lambda}$, and $u$ is a first nontrivial Laplace eigenfunction. Equivalently, if the first nonzero eigenvalue $\lambda_1$ of the Laplace-Beltrami operator attains the Lichnerowicz lower bound: $$\lambda_1 \geq n K, \qquad \text{for } \mathrm{Ric}_M \geq K>0,$$ with equality, then $M$ is isometric to the sphere of appropriate radius (Wu et al., 2012, Ketterer, 2014).

This rigidity statement extends to warped-product models for generalizations of the Hessian equation $\nabla^2 w + f(w)g = 0$, and also has Euclidean and hyperbolic analogs when $f(w)$ is constant or negative, respectively.

2. Obata-Type Rigidity in CR and Sub-Riemannian Settings

Obata-type rigidity has deep generalizations in geometric contexts involving non-integrable distributions and subelliptic operators. On a compact strictly pseudoconvex pseudohermitian manifold $(M^{2n+1}, \theta)$, define the horizontal bundle $H = \ker\theta$, almost complex structure $J$, Webster metric $g$, sub-Laplacian $\Delta_b$, and pseudohermitian torsion $A$. Under a divergence-free torsion condition $\nabla^{*}A = 0$ and a Lichnerowicz-type curvature bound $\mathrm{Ric}(X, X) + 4A(X, JX) \geq k_0 g(X, X)$, one has (Ivanov et al., 2012): $\lambda_1 \geq \frac{n}{n+1} k_0,$ and, up to homothety, equality $\lambda_1 = 2n$ characterizes $(M, \theta)$ as the standard Sasakian sphere.

Extensions to sub-Riemannian H-type manifolds with transverse symmetries yield rigidity to 1-Sasakian and 3-Sasakian spheres when the first eigenvalue of the sub-Laplacian attains an explicit sharp bound depending on the dimension and the rank of the vertical bundle (Baudoin et al., 2014).

Weighted versions on Sasakian and CR manifolds introduce Witten-type Laplacians and Bakry–Émery Ricci tensors; the sharp eigenvalue bound involves the weight oscillation, and equality again forces the structure to be CR-isometric to the sphere and the weight to be constant (Chang et al., 2019, Wu, 2019).

3. Obata-Type Theorems for Metric Measure Spaces and Stratified Spaces

In non-smooth settings, such as synthetic metric measure spaces $(X, d, m)$ with curvature-dimension conditions RCD$^*(K, N)$, Obata's rigidity persists. If there exists a nontrivial test function $u$ solving

$L^X u = -\frac{KN}{N-1} u,$

then $X$ must be a spherical suspension, and $u$ is a model cosine function. Moreover, the equivalence of lower Hessian bounds $H[u] \geq K$ with $K$-convexity is established (Ketterer, 2014).

For stratified spaces $(X^n, g)$ (with positive Ricci on the regular set and controlled cone angles), the equality in the spectral gap $\lambda_1(\Delta) = n$ implies that $X$ is a spherical suspension of a lower-dimensional stratified space (Mondello, 2015), with correlates for the Yamabe problem and conformal Einstein deformations.

4. Obata-Type Rigidity on Manifolds With Boundary and Robin Conditions

Rigidity results have been established for manifolds with boundary via overdetermined boundary value problems. For instance, on $(\Omega^{n+1}, g)$ smooth, with compact boundary and $f$ solving

$$\nabla^2 f \pm f g = 0 \quad \text{in } \Omega, \qquad f_{\nu} = c f \quad \text{on } \partial\Omega,$$

with $c$ determined by geometric parameters, if additional curvature and mean curvature bounds hold, then $(\Omega, g)$ is necessarily isometric to a (hyperbolic or spherical) geodesic ball of explicit radius, and $f$ is determined by the corresponding radial eigenfunction. The boundary being totally umbilic or achieving sharp diameter is equivalent to attaining rigidity (Liu et al., 2024, Almaraz et al., 2017).

In static manifolds with boundary, the presence of a positive static potential $V$ implies the “Obata–Robin” PDE system, and only two cases are possible: Ricci-flat geometry with totally geodesic boundary and constant static potential, or a warped-product structure over a Ricci-flat cross-section with exponential warping (Sheng et al., 5 Jan 2026). For boundary geometries not matching the Obata-type model, the scalar curvature–mean curvature map is locally surjective, contrasting with the rigidity of the closed-manifold setting.

5. Rigidity for Kähler and Calabi Geometries

Obata-type rigidity admits strong analogues in Kähler geometry. For a complete Kähler manifold $(M^{2n}, g, J)$ admitting a smooth function $f$ with $J$-linear Hessian and at most two distinct eigenvalues, with the gradient always a Hessian-eigenvector, the only possibilities are Calabi metrics on line bundles over totally geodesic complex submanifolds. If the bundle is flat, $M$ is biholomorphic to the total space of a trivial bundle with the standard Calabi metric; otherwise, metrics arise from line bundles whose Chern curvature is proportional to the base Kähler–Einstein metric (Ginoux et al., 2020).

Doubly-warped product Kähler manifolds (with prescribed warping functions and Sasakian transversal structures) provide the complete classification under Obata-type Hessian conditions plus Einstein metrics: all solutions are warped-product models parametrized by explicit ODEs governing the warping, with the curvature sign distinguishing projective, hyperbolic, or flat Calabi geometries (Ginoux et al., 2020).

6. Generalization, Spectral Rigidity, and Further Extensions

Spectral versions of Obata rigidity extend to higher eigenvalues, mixed boundary eigenvalue problems, and generalized Hessian equations with arbitrary nonlinearity. For instance, in Riemannian geometry, the equality case for the first eigenvalue under Ricci lower bounds or for specific Robin boundary conditions rigidly classifies the underlying manifold or domain.

In sub-Riemannian, CR, or metric measure settings, the characterization of model spaces (e.g., spheres, suspensions, Hopf fibrations) in the equality case fundamentally relies on the existence of extremal functions solving the relevant overdetermined PDE system, typically yielding warped-product or symmetric metrics.

Rigidity theorems are proved using Bochner-type identities, gradient flow splitting, comparison arguments, and eigenfunction characterizations. The analysis is local-to-global, often leveraging properties of the Hessian, curvature, and boundary geometry, and explicit integral identities.

Obata-type rigidity remains a central paradigm in spectral geometry, curvature analysis, and the classification of invariant geometric structures across a wide array of smooth and singular contexts.