Sur les périodes des courbes et la rigidité des systèmes liés
Sur les périodes des courbes et la rigidité des systèmes liés
Marie-Rose Simone, Paris, 1898
Communication à la Société Mathématique de France
The recent work of M. Poincaré on Analysis Situs has given us a powerful new language for geometry. We may now speak of cycles and boundaries, of periods and homology, with precision. I wish to show how these ideas illuminate a question in the theory of algebraic surfaces that has not yet received proper attention.
The question is this: when can a curve move freely upon a surface?
§1. The local calculation.
The geometer of surfaces will count the infinitesimal freedoms of a curve C lying upon a surface S. At each point, one asks: in how many directions may C be displaced? The answer is a number — let us call it n — computable from the local properties of C and S.
This is well understood. The difficulty arises when we ask whether these infinitesimal freedoms integrate to actual motion. Can we move C to a new position, or do the local freedoms cancel against one another?
§2. A lesson from the body.
I will describe two experiments that any reader may perform.
First experiment. Stand and clasp your hands before your chest. Your arms form a closed curve:
right shoulder → elbow → hand → left hand → elbow → shoulder → chest → right shoulder
This curve bounds a disk — an imaginary surface passing through the space between your arms and your chest, the circle enclosed by your embrace. The disk is not flesh, but it is interior to your system. It lies within the boundary of your being.
Now move your clasped hands freely. You will find that you can: up, down, forward, in circles. The local freedoms integrate to global motion. The curve that bounds an interior disk is free.
Second experiment. Stand with both feet planted on the earth. Your body again forms a closed curve:
left foot → earth → right foot → hip → pelvis → hip → left foot
This curve does NOT bound a disk interior to your system. To fill it, you would need the earth itself — and the earth is not yours. It is exterior. It is the external world, the observer, the constraint.
Now try to walk without lifting a foot. You cannot. You may wobble, oscillate, sway — but your center does not travel. The local freedoms do not integrate to global motion. The curve that passes through the exterior is rigid.
§3. The theorem of periods.
Let M be the configuration space of a system. A 1-cycle γ in M is interior if it bounds a 2-chain lying entirely within M. It is exterior if it passes through a region external to M — through a constraint, an observer, an attachment to the world.
Let ω be a closed 1-form representing infinitesimal displacement.
Proposition. If γ is interior (γ ~ 0 in M), then ∫ω = 0 automatically, by Stokes’ theorem. No constraint is imposed.
If γ is exterior (γ ≁ 0 in M, but passes through a fixed external region E), then we must have ∫ω = 0 by the rigidity of E. This is a constraint imposed from outside.
Corollary. Let b₁^{ext} denote the number of independent exterior cycles. The true freedom of the system is:
(local freedom) − b₁^{ext}
The local calculation overcounts by exactly the number of independent attachments to the external world.
§4. The boundary of being.
What determines whether a cycle is interior or exterior?
It is not a question of flesh. The disk between my clasped arms passes through air, not through my body. Yet it is interior to my system — it lies within the space I occupy, the region I can sweep with my limbs, the boundary of my energetic presence.
The earth, by contrast, is definitively exterior. It is not mine. I am coupled to it, but it is not part of my manifold.
The distinction is: what can I fill with my own substance? Here “substance” means not matter, but presence. The interior is what belongs to me. The exterior is the world.
§5. Application to curves on surfaces.
Consider now a curve C on a surface S. If C passes through a point P that is held fixed — pinned to an external constraint — then C forms an exterior cycle. The period around this cycle must vanish. This imposes a condition invisible to local calculation.
More generally: let C be linked with other curves, or pass through singular points, or wrap around handles of S in complicated ways. Each passage through an external constraint creates an exterior cycle. Each exterior cycle kills one degree of freedom.
The local geometer, counting infinitesimal directions, will overcount. The true freedom requires understanding which cycles escape the system and which remain interior.
§6. A warning for future geometers.
I expect that the theory of surfaces will advance rapidly in the coming years. Geometers will classify surfaces, study the motion of curves upon them, compute dimensions of families.
When they do, let them remember the woman standing on the earth. Let them ask: does this curve pass through an external constraint? Does it form an exterior cycle? Have I counted only interior freedoms, or have I mistaken an exterior rigidity for an interior flexibility?
The local calculation is necessary. It is not sufficient.
Remerciements. I thank M. Poincaré for the theory of homology, and my teacher of the bodily arts, who showed me that energy flows in circuits, and that circuits may close through the self or through the world.
Remarque finale. C’est par le corps que nous comprenons les périodes.
It is through the body that we understand periods.