Non-Holonomic Tetrads in Diff. Geometry

• Non-Holonomic Tetrads are local frames of vector fields characterized by noncommuting components, essential for modeling non-integrable distributions.
• They employ Cartan's structure equations and normalization techniques to reveal torsion, curvature, and differential invariants critical in geometric analysis.
• Their applications span sub-Riemannian geometry, geometric control theory, and general relativity, offering a robust framework for classifying complex geometric structures.

A non-holonomic tetrad is a local frame of vector fields on a differentiable manifold whose commutators do not necessarily vanish, providing a rigorous and flexible alternative to coordinate (holonomic) frames. These frames, and their dual coframes, underpin the geometric analysis of non-holonomic distributions, especially in the context of sub-Riemannian geometry and Cartan’s method of moving frames. The structure and classification of such tetrads are central to the study of non-integrable distributions and their invariants, particularly in the strongly non-holonomic case originally analyzed by Élie Cartan (Koiller et al., 2011, &&&1&&&).

1. Non-Holonomic Distributions and Tetrads

Let $Q^n$ be an $n$-dimensional smooth manifold equipped with a rank-$m$ distribution $E \subset TQ$ ($m < n$). The codistribution (Pfaffian system) $I = \mathrm{Ann}(E) \subset \Omega^1(Q)$ is generated by $n-m$ independent 1-forms $\omega^{m+1}, \dots, \omega^n$. A distribution $E$ is non-holonomic if it is non-integrable, which can be detected by the growth of its derived flag: $E^1 = E, \quad E^{k+1} = E^k + [E^k, E^k]$ If $E^k$ eventually equals $TQ$, the system is maximally non-integrable.

A local frame $\{e_a(x)\}$ is called a non-holonomic tetrad if its sections do not commute: $[e_a, e_b] = C^c{}_{ab}(x) e_c$ where $C^c{}_{ab}$ are the nontrivial structure functions of the distribution (Santos, 2017). These frames, together with their dual coframes $\{\theta^a(x)\}$ ($\theta^a(e_b) = \delta^a_b$), form the basis for non-coordinate analysis and exhibit nontrivial torsion and curvature properties.

2. Strongly Non-Holonomic Distributions Following Cartan

In Cartan’s framework, a distribution is strongly non-holonomic if the first derived Pfaffian system vanishes: $$I^{(1)} = \{ \theta \in I \mid d\theta \equiv 0 \mod I \} = 0$$ Under this condition, no nonzero $1$-form in $I$ has an exterior derivative contained in the ideal generated by $I$. Consequently, the orthogonal complement $F = E^\perp$ (with respect to any chosen metric) becomes intrinsic. Cartan showed that under coframe changes preserving the sub-Riemannian metric, the complement $F$ is canonically defined, as the “off-diagonal” block in the coframe transformation vanishes ($B = 0$). This property singles out strongly non-holonomic distributions as a rich geometric setting for intrinsic analysis (Koiller et al., 2011).

3. Cartan's Structure Equations for Non-Holonomic Tetrads

For a Riemannian or pseudo-Riemannian metric $g$ on $Q$, one constructs local orthonormal frames $\{e_i\}_{i=1}^m \subset E$ and $\{e_\alpha\}_{\alpha=m+1}^n \subset F$. The corresponding dual coframe $\{\omega^I\}$ satisfies

$$\omega^i(e_j) = \delta^i_j, \quad \omega^a(e_b) = \delta^a_b, \quad \omega^i(e_\alpha) = \omega^a(e_j) = 0$$

Cartan's structure equations, using the connection $1$-forms $\omega^I{}_J$, are

$$d\omega^I + \sum_J \omega^I{}_J \wedge \omega^J = T^I$$

$$d\omega^I{}_J + \sum_K \omega^I{}_K \wedge \omega^K{}_J = R^I{}_J$$

where $T^I$ and $R^I{}_J$ are torsion and curvature $2$-forms, respectively (Santos, 2017).

When restricting the connection to $E$, nonzero torsion appears: $$D_X e_j = \sum_{i=1}^m \omega^i{}_j(X) e_i, \quad X \in \Gamma(TQ)$$ The first structure equation on $E$ becomes

$$d\omega^i + \sum_{j=1}^m \omega^i{}_j \wedge \omega^j = T^i$$

with

$$T^i = \frac{1}{2} T^i{}_{jk} \omega^j \wedge \omega^k + S^i{}_{j\alpha} \omega^j \wedge \omega^\alpha + U^i{}_{\alpha\beta} \omega^\alpha \wedge \omega^\beta$$

The curvature of the restricted connection pulls back from the ambient connection: $$R^i{}_j = d\omega^i{}_j + \sum_{k=1}^m \omega^i{}_k \wedge \omega^k{}_j$$ (Koiller et al., 2011, Santos, 2017).

4. Normalization and Differential Invariants

Coframe changes preserving the sub-Riemannian metric act as block matrices. In the strongly non-holonomic case, the vanishing of the $B$ block yields a canonical complement and reduces the freedom to $O(m) \times O(n-m)$. Further normalization imposes that the pure-horizontal torsion coefficients become symmetric: $T^i{}_{jk} = T^i{}_{kj}$ Connection forms $\omega^i{}_j$ can be adjusted by semi-basic terms to achieve this symmetrization: $$\omega^i{}_j \mapsto \omega^i{}_j + P^i{}_{jk} \omega^k$$ with $P^i{}_{kl} = \frac{1}{2} (V^i{}_{kl} - Y^i{}_{lk})$, where $Y^i{}_{jk}$ and $V^i{}_{jk}$ are pre-normalization coefficients.

The normalized structure equations take the form: $$d\omega^i = -\omega^i{}_j \wedge \omega^j + T^i{}_{kl} \omega^k \wedge \omega^l + S^i{}_{k\alpha} \omega^k \wedge \omega^\alpha$$ with uniquely determined $T^i{}_{kl} = T^i{}_{lk}$. On the complement $F$, Gram–Schmidt orthonormalization further fixes the metric structure: $C^a{}_{ij} C^b{}_{ij} = S^{ab}$ The residual undetermined functions

$\{ T^i{}_{kl},\, S^i{}_{k\alpha},\, C^a{}_{ij} \}$

constitute differential invariants and classify the local strongly non-holonomic geometry (Koiller et al., 2011).

5. Commutation Relations and Structure Functions

Non-holonomic tetrads feature nontrivial structure functions $C^c{}_{ab}$, encapsulating the failure of the frame fields to commute. Explicitly,

$$C^c{}_{ab} = e^c{}_\nu \left( e_a{}^\mu \partial_\mu e_b{}^\nu - e_b{}^\mu \partial_\mu e_a{}^\nu \right)$$

or, equivalently, in terms of the dual coframe,

$$C^c{}_{ab} = e_a{}^\mu e_b{}^\nu \left( \partial_\mu e^c{}_\nu - \partial_\nu e^c{}_\mu \right)$$

These structure functions enter directly into the Cartan structure equations, and their symmetries and nonvanishing components reflect the essential non-holonomy of the frame (Santos, 2017).

6. Applications and Computation in Geometric Contexts

Non-holonomic tetrads are foundational in sub-Riemannian geometry, geometric control theory, and the analysis of non-integrable distributions. In General Relativity, tetrads (or vierbeins, in four dimensions) allow analyses in locally orthonormal frames, streamlining the calculation of curvature tensors as compared to coordinate-based methods.

A standard example is the construction of an orthonormal tetrad in Schwarzschild spacetime: $$ds^2 = -f(r)\,dt^2 + f(r)^{-1} dr^2 + r^2 (d\theta^2 + \sin^2\theta\,d\phi^2), \quad f(r) = 1 - \frac{2M}{r}$$ with coframe

$$\theta^0 = \sqrt{f}\,dt, \quad \theta^1 = f^{-1/2}\,dr, \quad \theta^2 = r\,d\theta, \quad \theta^3 = r\sin\theta\,d\phi$$

Exterior derivatives $d\theta^a$ then determine the connection 1-forms and, via Cartan's equations, the curvature. This approach replaces the computation of Christoffel symbols with algebraic calculations in the chosen tetrad, improving transparency and tractability (Santos, 2017).

7. Classification and Cartan's Method of Equivalence

The classification of local models of strongly non-holonomic structures proceeds by examining the fundamental invariants:

• Symmetric torsion coefficients $T^i{}_{kl}$
• Mixed torsion $S^i{}_{k\alpha}$
• Curvature-type invariants $R^i{}_{jkl}$

These tensors transform under the residual structure group $O(m) \times O(n-m)$. Cartan’s method of equivalence involves building a complete set of invariants through successive exterior derivatives of the normalized structure equations and addressing the integrability conditions, systematically reducing the remaining gauge freedom. This approach underlies the local and global classification of non-holonomic geometric structures (Koiller et al., 2011).

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Table: Summary of Key Objects in Non-Holonomic Tetrad Theory

Object	Definition/Role	Reference Section
$E \subset TQ$	Distribution (subbundle of tangent bundle)	1, 2
$I = \mathrm{Ann}(E)$	Pfaffian system (annihilator of $E$)	1, 2
$C^c{}_{ab}$	Structure functions (non-commutation of frame)	5
$T^i{}_{kl}$	Symmetric torsion coefficients (invariants)	4, 7
$S^i{}_{k\alpha}$	Mixed torsion (invariant)	4, 7
$R^i{}_{jkl}$	Curvature from restricted connection	3, 7

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Non-holonomic tetrads offer a powerful formalism for encoding the local geometry of non-integrable distributions, supporting both the explicit calculation of geometric objects (torsion, curvature) and the systematic classification and analysis of their invariants through Cartan’s moving frame method (Koiller et al., 2011, Santos, 2017).