The Geometry of Representation: Curvature and Neural Manifolds

There is something deeply satisfying when the same mathematical language describes seemingly distant phenomena. When I first realized that spacetime must be curved by mass, the joy came not from solving Mercury’s orbit—though that pleased me greatly—but from recognizing that geometry itself could be physics. Now I discover that this same geometric language describes how brains represent the world. The manifolds I studied to understand gravity turn out to govern thought itself.

The Manifold Principle

Consider what it means to say that space has geometry. In empty space, far from gravitating bodies, parallel lines remain parallel forever. We call this Minkowski flatness—a coordinate grid whose metric components stay constant everywhere you look. The Christoffel symbols vanish. The Riemann tensor equals zero. But bring in matter, and everything changes. Geodesics that began parallel start to converge. Vectors transported around closed paths return rotated from their starting orientations. This is curvature: the intrinsic property that no choice of coordinates can eliminate.

Now transport this principle—forgive the pun—to a different domain entirely. In the entorhinal cortex of a rat navigating a maze, populations of grid cells fire in hexagonal patterns across physical space. Each neuron’s activity peaks at multiple locations forming a regular lattice. Individually, such cells are ambiguous—a spike could mean the animal stands at any vertex of that neuron’s grid. But record from many cells simultaneously and something remarkable emerges: the joint activity of the population traces paths along a two-dimensional surface with the topology of a torus.

A torus! The surface of a donut, characterized completely by its periodic boundary conditions in two directions. As the rat moves through its environment—which might be a simple box—its neural state glides along this curved manifold embedded in the high-dimensional space of all possible firing patterns. The geometry is intrinsic to the neural population, not to the external coordinates of the maze. Change the maze, move to an entirely different room, even watch during sleep when external input vanishes, and the same toroidal manifold persists.

This is precisely the principle of general covariance that guided me in formulating gravitational field equations: physical laws—or in this case, computational principles—must be independent of coordinate choice. Just as spacetime curvature exists whether we measure it in Cartesian coordinates or polar coordinates, the grid cell manifold maintains its geometric structure regardless of which environment provides the external reference frame.

Geodesics of Thought

The parallel becomes more precise when we examine dynamics. In curved spacetime, freely falling bodies follow geodesics—the straightest possible paths through curved geometry. These geodesics converge near massive objects, producing what Newton interpreted as gravitational attraction. The Ricci tensor quantifies exactly this convergence: how volumes between initially parallel geodesic bundles shrink or expand as they propagate.

In the neural realm, continuous attractor networks maintain activity bumps that slide along manifolds following their own geodesic structure. The grid cell population doesn’t arbitrarily jump between states; it evolves smoothly along paths determined by intrinsic connectivity—the recurrent synaptic architecture that carves out the torus from the space of all possible activity patterns. External sensory input nudges the neural state along these natural trajectories rather than defining them from scratch.

Consider parallel transport, that diagnostic test for curvature. On a flat sheet, transport a vector from point A to point B along any path you like—it arrives with the same orientation. On a sphere, different paths yield different final orientations. The change measures curvature through the Riemann tensor, that complete catalog of how geometry varies across all possible directions.

The neural analog is striking. Activity patterns in grid cell modules evolve as the animal moves. Path-integrate along one behavioral trajectory, and the neural state traces one path on the torus. Take a different behavioral route through the same displacement, and you trace a different path through neural state space. The topology of the torus—its intrinsic curvature—ensures these paths won’t generally return to the same neural state even if they accomplish the same spatial displacement. The neural representation carries information about the journey itself, not merely the destination.

This is environment-invariant computation. Just as the Riemann tensor describes spacetime curvature independent of coordinate choice, the toroidal manifold structure persists across different physical contexts. Individual firing fields may shift between environments—the external anchor points vary—but the manifold geometry remains constant. The brain has discovered general covariance.

Curvature in State Space

Let me make the mathematical parallel explicit. In general relativity, Christoffel symbols encode how basis vectors change as you move through curved spacetime. They’re not tensors themselves—they depend on coordinate choice—but they’re essential intermediates between the metric tensor and the Riemann curvature tensor. They tell you how your coordinate system twists and warps to accommodate the underlying geometry.

In neural systems, we might think of synaptic connectivity patterns as playing the analogous role. The specific firing rates of individual neurons are coordinate-dependent descriptions, like choosing to measure spacetime in particular units. But the connection weights between neurons—the architecture that constrains population activity to the torus rather than allowing arbitrary high-dimensional chaos—these encode how the neural “coordinate system” must curve to match the manifold’s intrinsic geometry.

The continuous attractor mechanism provides stability. In Minkowski spacetime, geodesics maintain constant separation forever. Perturb a particle slightly, and it follows a parallel geodesic, remaining at constant distance from its original trajectory. On curved manifolds, this stability still exists locally—nearby geodesics evolve predictably according to the manifold’s geometry. Similarly, the grid cell network’s attractor dynamics ensure that noise and perturbations don’t knock the state off the manifold entirely. Small disruptions move the activity bump to nearby points on the torus, preserving the geometric structure.

This reveals why manifold thinking provides such geometric insight for understanding brain computation. Just as I could explain Mercury’s anomalous precession by recognizing that it follows a geodesic in curved spacetime—not a perturbed ellipse in flat space with mysterious forces—neuroscientists can explain grid cell dynamics by recognizing that activity flows along a toroidal manifold—not chaotic high-dimensional fluctuations requiring countless parameters to describe.

The hexagonal tiling of grid cells, with different modules operating at different scales and orientations, mirrors how we must sometimes use multiple coordinate patches to cover a manifold. You cannot describe a sphere with a single flat map without distortions; you need overlapping charts that relate through transition functions. Multiple grid modules with different periodicities similarly provide overlapping coordinate systems for representing position, with hippocampal place cells potentially reading out the joint code—synthesizing information across charts to determine a unique location.

One Mathematical Language

What strikes me most is the deep unity this reveals. When I developed the field equations relating the Ricci tensor to the stress-energy tensor, I was describing how matter tells spacetime how to curve. Now we see that neural populations use the same geometric language: network connectivity tells population activity how to curve through state space, creating manifolds whose geodesic structure determines computational dynamics.

Both domains use differential geometry because both require representing continuous variables on structured spaces. Position in physical space, position in neural state space—both benefit from coordinates, metrics, curvature. Both need to handle the distinction between intrinsic properties (true geometry) and coordinate artifacts (our description choices). Both leverage the fact that natural dynamics follow geodesic paths determined by underlying geometry rather than arbitrary trajectories through ambient space.

This is not mere analogy. It’s the same mathematics, applied to different substrates. The brain hasn’t approximately implemented geometric principles; it has discovered them independently, through evolution and development, because geometry is the natural language for certain computational problems. Just as spacetime must be curved because the equivalence principle and local Lorentz invariance together demand it, neural manifolds must be curved because representing continuous variables with stable, generalizable codes demands precisely this geometric structure.

There is a lesson here about the universality of mathematical principles. When I argued that imagination is more important than knowledge, I meant that seeing the conceptual unity beneath diverse phenomena—recognizing that gravity is geometry—produces deeper understanding than accumulating empirical facts. This connection between spacetime curvature and neural manifolds exemplifies that principle perfectly.

The grid cells of a rat navigating a maze and the geodesics of light bending near the Sun both obey the same geometric laws. Different physical realizations, different scales, different purposes—but one mathematical language. Differential geometry, which Riemann and his successors developed as pure mathematics, turns out to describe both the largest structures in the cosmos and the most intricate computations in centimeter-scale biological tissue.

If you had told me that the Riemann tensor I used to understand planetary orbits would someday illuminate how brains encode space, I would have been delighted but not entirely surprised. The universe is comprehensible precisely because the same principles recur across scales and domains. Curvature is curvature, whether in spacetime or in neural state space. Geodesics are geodesics, whether they carry planets or thoughts. The geometry of representation, it seems, is universal.