SUBLOR is a research project funded by the European Union under the Horizon Europe programme's Marie Skłodowska-Curie Actions Postdoctoral Fellowships, grant agreement No. 101282277.

The project

The SUBLOR project is devoted to sub-Lorentzian geometry, the non-holonomic analogue of Lorentzian geometry and the Lorentzian counterpart of sub-Riemannian geometry.

A sub-Lorentzian space is a manifold \( M \) together with a family of admissible directions \( \mathcal{D} \subseteq \mathrm{T}(M) \), a Lorentzian inner product \( \langle \cdot, \cdot \rangle \) on those directions, and a choice of time orientation in \( \mathcal{D} \). Typically the number of independent directions available is smaller than the dimension of \( M \), so motion is constrained: not every tangent direction in \( \mathrm{T}_x(M) \) is available.

A curve \( \gamma \) is called horizontal when its velocity stays inside \( \mathcal{D} \). It is causal if \( \langle \dot\gamma(t), \dot\gamma(t)\rangle_{\gamma(t)} \le 0 \), chronological if \( \langle \dot\gamma(t), \dot\gamma(t)\rangle_{\gamma(t)} < 0 \), and it is future-directed when it points consistently along the chosen time orientation. The length of a causal curve is given by the usual Lorentzian formula \[ L(\gamma) := \int_0^T \sqrt{-\langle \dot\gamma(t), \dot\gamma(t) \rangle_{\gamma(t)}} \, \mathrm{d}t. \]

The time-separation function \( \tau(x,y) \) is obtained by taking the supremum of these lengths among all horizontal future-directed causal curves from \( x \) to \( y \): \[ \tau(x, y) := \sup\left\{ L(\gamma) \;\middle|\; \gamma \text{ is horizontal, future-directed causal, from } x \text{ to } y \right\}. \] The associated causal and chronological futures and pasts, denoted by \( J^\pm(x) \) and \( I^\pm(x) \), are defined as in Lorentzian geometry, but using only horizontal causal or timelike curves.

This setup models constrained spacetimes where motion is limited to a subbundle of directions, analogous to sub-Riemannian geometry but in a Lorentzian setting. The figure above illustrates how a causal curve \( \gamma \) must remain tangent to \( \mathcal{D} \), with its velocity \( \dot{\gamma}(t) \) lying in the admissible future-directed cone. Every Lorentzian manifold is a sub-Lorentzian manifold by taking \( \mathcal{D} = \mathrm{T}(M) \), so sub-Lorentzian geometry generalises Lorentzian geometry.

While recent developments have extended Lorentzian geometry to the non-smooth setting and sub-Riemannian geometry has become a well-established and active area of research, sub-Lorentzian geometry remains largely underexplored. The project SUBLOR, mentored by Michael Kunzinger, will address this gap by:

1. Integrating sub-Lorentzian structures into the framework of Lorentzian length spaces and metric spacetimes.
2. Characterising the infinitesimal geometry of sub-Lorentzian manifolds via timelike metric tangents.
3. Developing curvature notions adapted to sub-Lorentzian geometry, including optimal-transport-based conditions and Hamiltonian curvature.

If you would like to hear more about SUBLOR, discuss related research, explore possible collaborations, or get involved, do get in touch!

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