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"The object obtained by associating a space of any number of dimensions with an affine connection is called an affine space."?
Step-by-step mathematics solution: "The object obtained by associating a space of any number of dimensions with an affine connection is called an affi
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1. What the question is really asking
The student has found two different “definitions’’ of the term affine space in the literature:
-
In Veblen’s Invariants of Differential Quadratic Forms (and in many old differential‑geometric texts) the phrase
“the object obtained by associating a space of any number of dimensions with an affine connection is called an affine space’’
is used.
-
In most modern textbooks the phrase “affine space’’ means a set on which a vector space acts freely and transitively (a torsor).
The student wants to know which of these two descriptions is the correct definition of an affine space.
2. Step‑by‑step analysis
2.1 The modern algebraic‑geometric definition
Let (V) be a real (or complex) vector space.
A (real) affine space modeled on (V) is a pair ((A,\tau)) where
- (A) is a non‑empty set,
- (\tau : V\times A \to A) is an action ((v,a)\mapsto a+v),
satisfying
| Property | Meaning |
|---|---|
| Free | If (a+v = a) for some (a\in A) then (v=0). |
| Transitive | For any (a,b\in A) there is a unique (v\in V) with (b = a+v). |
In other words, the difference of two points of (A) is a vector in (V), and adding a vector to a point moves you to another point.
Typical notation: if (p,q\in A) we write (\overrightarrow{pq}\in V) for the unique vector satisfying (q = p+\overrightarrow{pq}).
This definition does not involve any notion of smoothness, connection, curvature, etc. It is the definition used in linear algebra, projective geometry, computer graphics, etc.
2.2 “Affine space’’ in the differential‑geometric literature
In the early 20th‑century differential‑geometric tradition (Veblen, Weyl, Eisenhart, …) the word affine was used more loosely:
- An affine connection on a smooth manifold (M) is a rule (\nabla) that permits covariant differentiation of vector fields.
- If a smooth manifold (M) is equipped with a (generally non‑flat) affine connection, some authors called the pair ((M,\nabla)) an affine space.
- If, in addition, the connection is flat (vanishing curvature) and torsion‑free, then locally there exist coordinates in which all Christoffel symbols are zero; the manifold looks like ordinary Euclidean space. In that special case the same authors sometimes said the affine space is “flat’’ and could be regarded as Euclidean.
Thus Veblen’s sentence
“The object obtained by associating a space of any number of dimensions with an affine connection is called an affine space”
means “a smooth manifold together with a chosen affine connection’’. Modern terminology would call this an affine manifold (or an affine connection space) rather than an affine space.
2.3 Why the two meanings differ
| Aspect | Modern “affine space’’ (torsor) | Veblen’s “affine space’’ (affine connection) |
|---|---|---|
| Underlying set | Any set with a free, transitive (V)-action | A smooth manifold (M) |
| Extra structure | Only the action of a vector space | A connection (\nabla) (covariant derivative) |
| Curvature / torsion | Not defined | Defined via the connection; flat ↔ Euclidean |
| Typical notation | (A), (\overrightarrow{pq}) | ((M,\nabla)) |
| Modern name (if you keep Veblen’s meaning) | – | affine manifold (or manifold with affine connection) |
Both uses are historically correct, but they refer to different mathematical objects. In contemporary mathematics the word “affine space’’ is almost always reserved for the torsor definition. When a manifold equipped with a connection is meant, authors say “affine manifold’’ or “manifold with an affine connection’’ to avoid ambiguity.
2.4 Which definition should you adopt?
If you are reading a modern algebra/geometry text – use the torsor definition.
If you are reading an older differential‑geometric paper (Veblen, Eisenhart, etc.) – interpret “affine space’’ as “a smooth manifold equipped with an affine connection’’ (often called an affine manifold today).
Consequently, the “correct’’ definition depends on the context. In current standard terminology the first definition (torsor) is the accepted one; the second is an outdated usage that should be replaced by “affine manifold’’ when writing today.
3. Final answer
-
Standard modern definition:
An affine space is a set (A) on which a vector space (V) acts freely and transitively (a torsor for (V)). -
Veblen’s definition:
The phrase “affine space’’ is being used to mean a smooth manifold equipped with an affine connection; in modern language one would call this an affine manifold (or simply a manifold with an affine connection).
Thus the “correct’’ definition is the torsor one; Veblen’s usage is an older, non‑standard terminology that should be interpreted as “affine manifold’’ in present‑day language.
4. Common mistakes
| Mistake | Why it is wrong | How to avoid it |
|---|---|---|
| Confusing the two meanings and assuming a torsor automatically has a connection. | A torsor has no smooth structure, no notion of parallel transport, curvature, etc. | Always check whether the source talks about a manifold and a connection; if yes, they are using the older “affine space’’ meaning. |
| Calling any manifold with a connection an “affine space’’ without qualification. | Modern authors reserve “affine space’’ for the torsor; using the term loosely can lead to misunderstandings. | Use the term affine manifold or manifold with an affine connection for the differential‑geometric object. |
| Assuming flatness = Euclidean for a general affine connection. | Flatness (vanishing curvature) only guarantees locally Euclidean coordinates; globally the manifold may be a torus, a cylinder, etc. | Remember: flat ⇔ locally isomorphic to (\mathbb{R}^n); global topology can still be non‑trivial. |
| Neglecting torsion when the connection is not symmetric. | Veblen’s “affine connection’’ need not be torsion‑free; the presence of torsion changes the geometry. | State explicitly whether the connection is assumed symmetric (torsion‑free) or not. |
By keeping the two notions distinct and using the appropriate modern terminology, the confusion disappears.
Original question: "The object obtained by associating a space of any number of dimensions with an affine connection is called an affine space."? on Mathematics Stack Exchange, licensed CC BY-SA.