Extrinsic Curvature Components

• Extrinsic curvature components are defined via the second fundamental form, capturing how a submanifold bends within its ambient space.
• They are computed using classical differential geometry and statistical methods like PCA to extract measures such as principal and mean curvatures.
• These components are crucial in discrete geometric modeling, numerical relativity, and physical applications in higher codimension settings.

Extrinsic curvature components quantify how a submanifold is curved within its ambient manifold, capturing the deviation of its embedding from being totally geodesic. In differential geometry, these are encoded by the second fundamental form and related tensors, which decompose curvature into normal and tangent contributions. The representation, extraction, and approximation of extrinsic curvature components play a central role in geometric analysis, geometric modeling, numerical relativity, and the mathematical structure of physics in higher codimension and brane contexts.

1. Definition and Fundamental Forms

Given a smooth $n$-dimensional submanifold $M^n$ embedded in an ambient Riemannian ($\mathbb{E}^{n+1}$) or pseudo-Riemannian ($\mathbb{R}^{1,n+p}$) space, the extrinsic curvature is described through the second fundamental form $\operatorname{II}$, a symmetric bilinear form on the tangent spaces encoding the normal acceleration of tangent vector fields: $$\operatorname{II}_p(u, v) = \langle (\nabla_v^{\text{amb}} u)^\perp, n \rangle,$$ where $u, v \in T_p M$, $n$ is a choice of local unit normal, and $\nabla^{\text{amb}}$ is the ambient Levi-Civita connection.

Associated to $\operatorname{II}$ is the shape operator $S: T_p M \to T_p M$ given by

$\langle S(u), v \rangle = \operatorname{II}(u, v).$

The eigenvalues of $S$ are the principal curvatures $\kappa_1,\ldots,\kappa_n$; the corresponding eigenvectors are the principal directions. The mean curvature is $H = \operatorname{tr}(S) = \sum_{i=1}^n \kappa_i$ and the normal trace $\operatorname{tr}_{\perp}$ of higher fundamental forms encodes additional extrinsic geometric information, relevant notably in higher codimension contexts (Álvarez-Vizoso et al., 2018).

In higher codimension $(M^n \subset \mathbb{R}^{n+k})$, the second fundamental form becomes a vector-valued map $\operatorname{II}: T_p M \times T_p M \to N_p M$, with shape operators $S_n$ for each normal direction $n \in N_p M$, and the so-called "third fundamental form" is a fourth-rank tensor incorporating the action of $S_n$ in multiple normal directions (Álvarez-Vizoso et al., 2018).

2. Classical and Distributional Representations

Classical (Smooth) Case

Locally, the submanifold can be represented as the graph $$X(x) = (x, f(x)) \in \mathbb{R}^n \times \mathbb{R}$$, with $f(0) = 0$ and $\nabla f(0) = 0$ for some adapted orthonormal frame. The Hessian $\partial^2 f / \partial x^a \partial x^b$ at the point gives the components of the second fundamental form:

$$\operatorname{II}_p(e_a, e_b) = \frac{\partial^2 f}{\partial x^a \partial x^b} (0) \cdot n$$

for $a, b = 1, \ldots, n$.

The third fundamental form in the hypersurface case is $S^2$, with

$$\operatorname{III}(e_a, e_b) = \langle S^2 e_a, e_b \rangle, \qquad \operatorname{tr}_\perp \operatorname{III} = \sum_a \kappa_a^2.$$

Piecewise Flat (Discrete) Setting

In simplicial (piecewise-flat) geometry, extrinsic curvature is concentrated on codimension-1 faces (hinges) $h$. Given two adjacent $n$-simplices sharing $h$, with normals $\vec n_1$, $\vec n_2$ and a tangent direction orthogonal to $h$, the hinge (dihedral) angle $\phi_h$ is defined via: $$\phi_h = \arccos \langle \vec n_1, \vec v \rangle, \qquad \epsilon_h := \tfrac{\pi}{2} - \phi_h.$$ The discrete mean curvature at a vertex $v$ is then given as a weighted sum of hinge angles over a dual cell $V_v$: $$H_v = \frac{1}{|V_v|} \sum_{h \in \mathrm{Star}(v)} |h \cap V_v| \, \epsilon_h$$ with $|V_v|$ the $n$-volume of the dual cell and $|h \cap V_v|$ intrinsic $(n-1)$-volume (Conboye, 2023, Conboye, 2016).

Directed (hinge-orthogonal) curvature, which provides principal curvature components, is similarly an average of weighted hinge angles over a hinge region $V_h$, incorporating geometric weights dependent on the cell structure and the angles between normal directions (Conboye, 2023).

3. Extraction via Covariance Analysis and Integral Invariants

Extrinsic curvature components can be recovered from statistical analysis of local domains in the submanifold—most notably via principal component analysis (PCA) over small neighborhoods:

1. Place a small ball of radius $\varepsilon$ in the ambient space intersecting the manifold at $p$.
2. Compute the covariance matrix $C(\varepsilon)$ of the intersection domain points with respect to their barycenter:

$$C(\varepsilon) = \int_{D_p(\varepsilon)} [X - s(\varepsilon)] \otimes [X - s(\varepsilon)] d\mathrm{Vol}_M(X)$$

1. The eigenvalues $\lambda_i(\varepsilon)$ (tangent) admit an asymptotic expansion:

$$\lambda_i(\varepsilon) = V_n \varepsilon^{n+2} \left[ \frac{1}{n+2} + \frac{\varepsilon^2}{8(n+2)(n+4)} \big(2 \operatorname{tr} - H^2 - 4 \kappa_i H\big) + o(\varepsilon^2) \right]$$

where $V_n$ is the unit $n$-ball volume, $H$ is mean curvature, $\operatorname{tr} = \sum_{a=1}^n \kappa_a^2$ is the trace of $S^2$ (Álvarez-Vizoso et al., 2018).

From these expansions, one can invert explicit formulae to estimate the principal curvatures $\kappa_i$ and the mean and scalar curvatures in the $\varepsilon \to 0$ limit. Analogous constructions exist for higher-codimension submanifolds, where all components of the second fundamental form and the associated traces of the "third fundamental form" are encoded in the scaling behavior of the covariance spectrum (Álvarez-Vizoso et al., 2018).

The approach extends to integral invariants extracted from barycenter and volume, with direct multi-scale estimates for curvature, including scale-dependent smoothing and error assessment (Álvarez-Vizoso et al., 2018).

4. Discrete and Algorithmic Reconstruction

On piecewise-flat manifolds and in mesh-based geometry:

• Curvature "integrals" along paths crossing hinges are computed as sums of weighted hinge angles:

$$K_\gamma = \sum_{h \cap \gamma} \cos \theta_h \, \phi_h + O(\phi_h^3)$$

where $\theta_h$ measures the angle between the path and the hinge normal.

• The full curvature tensor on a simplex (e.g., triangle in 2D) is assembled from directed curvatures via polarization:

$$\alpha(u,v) = \frac{1}{2} [\kappa(u) + \kappa(v) - \kappa(u-v)]$$

enabling recovery of the local shape operator $K_{ab}$ (Conboye, 2023, Conboye, 2016).

All weights depend solely on intrinsic graph combinatorics, edge lengths, and the dual tessellation. Mesh refinement improves convergence ($O(\ell^2)$ errors for mean and directional curvatures in practical tests) and the method generalizes to arbitrary dimension and both Euclidean and non-Euclidean ambient geometries (Conboye, 2023, Conboye, 2016).

5. Extrinsic Curvature in General Relativity and Physical Contexts

In the context of $3$-geometry in general relativity, the extrinsic curvature tensor $K_{ij}$ encapsulates the rate of deformation of a spacelike hypersurface $\Sigma$ embedded in spacetime. Explicitly,

$K_{ij} = -\frac{1}{2} \mathcal{L}_n g_{ij}$

for $g_{ij}$ the induced metric and $n^\mu$ the unit normal. It appears directly in the Einstein constraint equations: $$R - K_{ij} K^{ij} + K^2 = 0\,\,\text{(Hamiltonian)},\quad \nabla_i K^{ij} - \nabla^j K = 0\,\,\text{(Momentum)}$$ The conformal decomposition $K^{ij} = K^{ij}_{TT} + \frac{1}{3} K g^{ij}$ separates gravitational wave (transverse-traceless) and expansion components. Scaling up the TT-part modifies the local wave "velocity", but due to the conformal factor's nonlinear adjustment, the physical wave energy density does not diverge; instead, the local spatial volume expands, and total energy grows as the region swells (Bai et al., 2012).

Such components also enter fundamentally in the analysis of brane-world models, quantum field theory on submanifolds, and the geometry of embedded surfaces, with direct physical consequences (e.g., induced potentials in compactifications and cosmological scenarios) (Langmann et al., 2011).

6. Clifford Algebras, Bilegendrian Structures, and Geometric Decomposition

The extrinsic curvature tensor can be encoded and reconstructed in geometric algebra frameworks. For immersed surfaces $S \to X$ in a $3$-manifold, the Gauss lift $S \to UX$ into the unit tangent bundle, combined with Clifford algebra and bilegendrian (simultaneously null for two compatible symplectic forms) structure, allows the second fundamental form $\operatorname{II}$ to be recovered as: $$\operatorname{II}(\xi, \eta) = \omega_2(V_e D\phi(\xi), D\phi(\eta))$$ where $V_e D\phi(\xi)$ represents the vertical part (derivative of normal) in the lifted frame and $\omega_2$ is the Clifford-derived symplectic structure. This yields the standard shape operator, principal curvatures, mean, and Gaussian curvature in terms of the Clifford algebra, operationalizing the decomposition directly in algebraic-geometric terms (Smith, 2023).

7. Summary Table: Key Quantities and Expansions

Quantity	Formula / Expansion	Context
Second Fundamental Form $\operatorname{II}_{ab}$	$$\partial^2 f / \partial x^a \partial x^b (0) \cdot n$$	Smooth embedding, local coordinates
Shape Operator S	$\langle S(u), v \rangle = \operatorname{II}(u, v)$	All settings
Principal Curvatures $\kappa_i$	Eigenvalues of $S$	All settings
Covariance Eigenspectrum	$$\lambda_i(\varepsilon) = V_n \varepsilon^{n+2} [\frac{1}{n+2} + \ldots ]$$	PCA/integral invariants (Álvarez-Vizoso et al., 2018)
Discrete Mean Curvature $H_v$	$$|V_v|^{-1} \sum_{h \in \mathrm{Star}(v)} |h \cap V_v| \epsilon_h$$	Piecewise-flat (Conboye, 2023)
General Relativity $K_{ij}$	$- \frac{1}{2} \mathcal{L}_n g_{ij}$	Cauchy data

This summary emphasizes the interplay between classical differential invariants, discrete geometric constructions, covariance-based statistical estimators, and algebraic formulations. The full structure of extrinsic curvature components is central to the intrinsic/extrinsic dichotomy in geometry, the analysis of embedded manifolds, and the formulation and solution of geometric-physical problems in both mathematics and physics.