Abstract
The phrase “fix the metric and the connection follows” quietly adds two demands. This paper counts that the two use up the freedom exactly, and measures what comes back when one of them is dropped. NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED.
First, demanding only that the metric be preserved does not pin it down. In $d$ dimensions there are $d^3$ unknowns against $d^2(d+1)/2$ equations, and the freedom left is $2$, $9$, $24$, $50$ at $d=2,3,4,5$ — matching the component count of a torsion tensor, $d^2(d-1)/2$, exactly (Section 2).
Second, adding “no torsion” uses the freedom up precisely. The unknowns and the rank agree at every dimension and the freedom left is $\mathbf{0}$ — the two demands are neither short nor spare (Section 3).
Third, the one thing so pinned is the Christoffel symbol. Its violation of metric compatibility is $5.55\times10^{-17}$ and its torsion is exactly zero (Section 4).
Fourth, this is the core. Same metric, same lengths, different straight. On the flat plane, choosing another admissible connection without touching the metric at all, two “straight” curves from the same point with the same velocity end up $1.186991$ apart (Section 5).
Fifth, within the admissible family the choices that leave the straight lines alone number exactly $\binom{d}{3}$. That is $\mathbf{0}$ at $d=2$, then $1$, $4$, $10$, $20$ — in two dimensions every admissible choice other than Levi-Civita moves the straight lines (Section 6).
Scope (what is not claimed)
NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED — the fundamental theorem of Riemannian geometry, torsion, Einstein–Cartan theory and Weyl connections are all standard. No verdict is given on which connection to adopt — this paper counts how many options there are and does not enter the question of which one physics should take. No census of the kinds of curvature is taken — that is the subject of Paper 91. Light and conformal structure are not treated — how a Weyl connection relates to conformal structure is touched on in Paper 111, and this paper does not enter there. Relation to Paper 95: 95 separated convention from fact — this paper draws that line across one object: the metric is given, and the connection is chosen. Relation to Papers 198 and 201: 198 measured where straight and shortest come apart, and 201 wrote that changing the signature turns the direction over — this paper writes what lies before both: that “straight” is fixed by a tool other than the metric. What is added is counting the freedom at four dimensions and checking it against the torsion component count, showing that the two demands fill the rank exactly, measuring how far the straight lines move while the metric is untouched, and confirming at five dimensions that the choices leaving them alone number $\binom{d}{3}$.
1. Notation and Position
A connection is the rule for comparing vectors at different points, written in components as $\Gamma^a{}_{bc}$. It is an object independent of the metric $g_{ab}$. Two demands are treated below. (i) Metric compatibility: parallel transport preserves lengths and angles, that is $\nabla_c g_{ab}=0$. (ii) No torsion: $\Gamma^a{}_{bc}=\Gamma^a{}_{cb}$. Every number below is computed here; ranks of linear systems are taken by singular value decomposition with a threshold of $10^{-9}$.
Position: Paper 95 separated “convention” from “fact”. This paper draws that line across a connection — the metric is what was laid down to measure with, and the connection is one further choice. How much freedom the two demands consume is then looked at as a number.
2. Preserving the Metric Does Not Pin It Down
The condition $\nabla_c g_{ab}=0$ is linear in $\Gamma$. The unknowns are the $d^3$ components of $\Gamma^a{}_{bc}$, and the equations number $d^2(d+1)/2$, since $c$ is free and $ab$ is symmetric.
Check 2.1: for a randomly drawn symmetric positive definite metric, measure the rank of the coefficient matrix.
| $d$ | Unknowns | Equations | Rank | Freedom left | $d^2(d-1)/2$ |
|---|---|---|---|---|---|
| $2$ | $8$ | $6$ | $6$ | $2$ | $2$ |
| $3$ | $27$ | $18$ | $18$ | $9$ | $9$ |
| $4$ | $64$ | $40$ | $40$ | $\mathbf{24}$ | $\mathbf{24}$ |
| $5$ | $125$ | $75$ | $75$ | $50$ | $50$ |
Observation 2.2 The freedom left matches the component count of a torsion tensor exactly. The demand that lengths be preserved says nothing at all about the torsion. In four dimensions, $24$ options remain untouched.
3. The Second Demand Uses the Freedom Up Exactly
Check 3.1: imposing $\Gamma^a{}_{bc}=\Gamma^a{}_{cb}$ reduces the unknowns to $d^2(d+1)/2$. Measure the rank against the same equations.
| $d$ | Unknowns | Equations | Rank | Freedom left |
|---|---|---|---|---|
| $2$ | $6$ | $6$ | $6$ | $\mathbf{0}$ |
| $3$ | $18$ | $18$ | $18$ | $\mathbf{0}$ |
| $4$ | $40$ | $40$ | $40$ | $\mathbf{0}$ |
| $5$ | $75$ | $75$ | $75$ | $\mathbf{0}$ |
Observation 3.2 At every dimension the unknowns and the equations agree in number, and the rank is full. The two demands are neither short nor spare — one fewer and a family would remain, one more and the solution would vanish. The fundamental theorem of Riemannian geometry can say “unique” because of this balance of counts.
4. And the One So Pinned Is the Christoffel Symbol
Check 4.1: in four dimensions, build the Christoffel formula from a random metric and its derivatives, and measure how far it satisfies the two demands.
| Quantity measured | Value |
|---|---|
| Largest $\nabla_c g_{ab}$ | $5.551\times10^{-17}$ |
| Largest torsion (antisymmetric part in the lower indices) | $\mathbf{0}$ exactly |
The first is floating-point rounding; the second follows from the formula being built symmetric.
5. The Core: Same Metric, Same Lengths, Different Straight
Take the flat plane with the identity as its metric. By the count of Section 2, the metric-preserving connections form a two-dimensional family. Levi-Civita is the one point of it with zero torsion, and the other points preserve lengths equally well.
Check 5.1: pick a member of the family that is not Levi-Civita, and advance an autoparallel from the same point with the same velocity over the same interval of the parameter.
| Connection | Endpoint |
|---|---|
| Levi-Civita (the ordinary straight line) | $(3.000000,\ 0.900000)$ |
| Another point of the family | $(2.427871,\ 1.940008)$ |
| Separation | $\mathbf{1.186991}$ |
Observation 5.2 The metric was never changed. So every curve has the same length and every pair of points the same distance. What changed is only which curves count as straight. Paper 198 measured where straight and shortest come apart — here they do not merely come apart: the “straight” side was handed in from outside the metric.
6. The Choices That Leave the Straight Lines Alone Number Exactly $\binom{d}{3}$
Only the part of $\Gamma$ symmetric in its two lower indices enters the autoparallel equation $\ddot x^a+\Gamma^a{}_{bc}\dot x^b\dot x^c=0$. So the choices that leave “straight” alone are exactly those whose symmetric part vanishes.
Check 6.1: measure the dimension of that subspace inside the admissible family.
| $d$ | Family dimension | Leaving the lines alone | $\binom{d}{3}$ |
|---|---|---|---|
| $2$ | $2$ | $\mathbf{0}$ | $0$ |
| $3$ | $9$ | $1$ | $1$ |
| $4$ | $24$ | $4$ | $4$ |
| $5$ | $50$ | $10$ | $10$ |
| $6$ | $90$ | $20$ | $20$ |
Check 6.2: in four dimensions, take one of each and check.
| Connection chosen | Size of torsion | Symmetric part | Shift of the endpoint |
|---|---|---|---|
| One whose symmetric part vanishes | $0.3846$ | $1.665\times10^{-16}$ | $\mathbf{0.000000000}$ |
| A generic member of the family | $0.2500$ | $0.5000$ | $2.223257376$ |
(Both violate metric compatibility only at the level of $10^{-16}$, that is, rounding.)
Observation 6.3 There are choices with nonzero torsion that move the straight lines not at all. “There is torsion” and “straight has changed” are separate claims.
Observation 6.4 And in two dimensions that number is $\mathbf{0}$. Because $\binom{2}{3}=0$: every admissible choice other than Levi-Civita moves the straight lines. Only from three dimensions on does it become possible to carry torsion without moving a single straight line. The room for choice itself depends on the dimension.
7. The Separator and the Fence
Observation 7.1 (the separator) What separates the metric from the connection is what each is a tool for comparing. The metric compares two vectors at the same point — it gives lengths and angles. The connection compares vectors at different points — it gives what counts as parallel. The second does not follow from the first: the $24$ of Section 2 is the size of what does not follow.
Observation 7.2 (the fence) The fence around “the metric determines the connection” is the extra demand that there be no torsion. Inside the fence it is unique, and the moment one steps out, $d^2(d-1)/2$ degrees of freedom return. And this fence is a choice, not a consequence of a theorem — Einstein–Cartan theory stands outside it with the same metric.
Observation 7.3 (dropping the other demand) Metric compatibility can be dropped instead — and then parallel transport changes lengths, giving a Weyl connection. This paper does not enter there: how a Weyl connection relates to conformal structure belongs to Paper 111.
8. The Census and the Close
What this paper checked:
- Preserving the metric does not pin it down (Check 2.1): freedom $2$, $9$, $24$, $50$, matching the torsion count.
- The two demands use the freedom up exactly (Check 3.1): full rank and zero freedom at every dimension.
- The one so pinned is Christoffel (Check 4.1): violation $5.55\times10^{-17}$, torsion exactly zero.
- Same metric, same lengths, different straight (Check 5.1): $1.186991$ apart on the flat plane.
- The quiet choices number $\binom{d}{3}$ (Check 6.1): $\mathbf{0}$ at $d=2$.
- Torsion without moving the lines (Check 6.2): shift $0.000000000$.
- The fence is the extra demand of no torsion (Observation 7.2).
References
- Levi-Civita, T. (1917). Nozione di parallelismo in una varietà qualunque. Rendiconti del Circolo Matematico di Palermo, 42.
- Weyl, H. (1918). Reine Infinitesimalgeometrie. Mathematische Zeitschrift, 2(3).
- Cartan, E. (1922). Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion. Comptes Rendus de l'Académie des Sciences, 174.
- Hehl, F. W., von der Heyde, P., Kerlick, G. D., Nester, J. M. (1976). General Relativity with Spin and Torsion. Reviews of Modern Physics, 48(3).
- Kobayashi, S., Nomizu, K. (1963). Foundations of Differential Geometry. Volume 1. New York: Interscience.
- Yamagishi, Y. (2026). Curvature Is Intrinsic [Paper 91]. Zenodo. 10.5281/zenodo.21758406
- Yamagishi, Y. (2026). Convention and Fact [Paper 95]. Zenodo. 10.5281/zenodo.21758621
- Yamagishi, Y. (2026). How Far Does Light Determine Geometry? [Paper 111]. Zenodo. 10.5281/zenodo.21961685
- Yamagishi, Y. (2026). “Straight” Does Not Mean “Shortest”, and There Are Two Fences [Paper 198]. Zenodo. 10.5281/zenodo.22107954
- Yamagishi, Y. (2026). The Two Hundredth Census [Paper 200]. Zenodo. 10.5281/zenodo.22108829
- Yamagishi, Y. (2026). “Complete” Is Four Separate Statements [Paper 201]. Zenodo. 10.5281/zenodo.22109052
Reference
Yamagishi, Y. (2026). A Connection Is Not Derived from the Metric: It Is Pinned Only Once Two Demands Are Chosen [Paper 202]. YOI: https://www.yamagishiyuuki.com/yoi/202
DOI: 10.5281/zenodo.22109349
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 202. This paper is a preprint and has not been peer reviewed.