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In GR an important, purely geometric equation is called Raychoudhuri's equation governing the behaviour of geodesic congruences which states that
$$\frac{d\theta}{d\tau} = - \frac{1}3 \theta^2 - \sigma^{ab}\sigma_{ab} + \omega^{ab}\omega_{ab} -R_{ab} u^a u^a$$
where ##R_{ab}## is the ricci tensor, ##\theta = \nabla_a u^a ## ##\omega_{ab}## and ##\sigma_{ab}## respectively are the trace, the antisymmetric and the symmetric bart of ##\nabla_a u_b## and u is the tangent vector field to the congruence. In other words this equation governs the spread in the geodesic congruence as a result of curvature.
Since this is a purely geometric result I wondered what this equation is called in the differential geometry literature, and I wondered if there were a formulation of this result that did not use index notation, but rather abstract notation.
$$\frac{d\theta}{d\tau} = - \frac{1}3 \theta^2 - \sigma^{ab}\sigma_{ab} + \omega^{ab}\omega_{ab} -R_{ab} u^a u^a$$
where ##R_{ab}## is the ricci tensor, ##\theta = \nabla_a u^a ## ##\omega_{ab}## and ##\sigma_{ab}## respectively are the trace, the antisymmetric and the symmetric bart of ##\nabla_a u_b## and u is the tangent vector field to the congruence. In other words this equation governs the spread in the geodesic congruence as a result of curvature.
Since this is a purely geometric result I wondered what this equation is called in the differential geometry literature, and I wondered if there were a formulation of this result that did not use index notation, but rather abstract notation.