Bonjour, je ne comprends pas bien la courbure d'une surface. Si je mets une fourmi sur une feuille de papier et qu'elle fait le tour, est-ce qu'elle verra que la feuille est courbe ? Merci d'avance pour votre réponse.
Gaussian curvature

In contrast to curves, which do not have intrinsic curvature, but do have extrinsic curvature (they only have a curvature given an embedding), surfaces can have intrinsic curvature, independent of an embedding. The Gaussian curvature, named after Carl Friedrich Gauss, is equal to the product of the principal curvatures, k k. It has a dimension of length and is positive for spheres, negative for one-sheet hyperboloids and zero for planes and cylinders. It determines whether a surface is locally convex (when it is positive) or locally saddle-shaped (when it is negative).

Gaussian curvature is an intrinsic property of the surface, meaning it does not depend on the particular embedding of the surface; intuitively, this means that ants living on the surface could determine the Gaussian curvature. For example, an ant living on a sphere could measure the sum of the interior angles of a triangle and determine that it was greater than 180 degrees, implying that the space it inhabited had positive curvature. On the other hand, an ant living on a cylinder would not detect any such departure from Euclidean geometry; in particular the ant could not detect that the two surfaces have different mean curvatures (see below), which is a purely extrinsic type of curvature.

Formally, Gaussian curvature only depends on the Riemannian metric of the surface. This is Gauss 's celebrated Theorema Egregium, which he found while concerned with geographic surveys and mapmaking.

An intrinsic definition of the Gaussian curvature at a point P is the following: imagine an ant which is tied to P with a short thread of length r. It runs around P while the thread is completely stretched and measures the length C (r) of one complete trip around P. If the surface were flat, the ant would find C (r) = 2π r. On curved surfaces, the formula for C (r) will be different, and the Gaussian curvature K at the point P can be computed by the Bertrand–Diguet–Puiseux theorem as

The integral of the Gaussian curvature over the whole surface is closely related to the surface's Euler characteristic; see the Gauss–Bonnet theorem.

The discrete analog of curvature, corresponding to curvature being concentrated at a point and particularly useful for polyhedra, is the (angular) defect; the analog for the Gauss–Bonnet theorem is Descartes' theorem on total angular defect.

Because (Gaussian) curvature can be defined without reference to an embedding space, it is not necessary that a surface be embedded in a higher-dimensional space in order to be curved. Such an intrinsically curved two-dimensional surface is a simple example of a Riemannian manifold.
Surfaces

La courbure des courbes tracées sur une surface est l'outil principal pour définir et étudier la courbure de la surface.
### Query ###
Bonjour, je ne comprends pas bien la courbure d'une surface. Si je mets une fourmi sur une feuille de papier et qu'elle fait le tour, est-ce qu'elle verra que la feuille est courbe ? Merci d'avance pour votre réponse.

### Main Text ###
Gaussian curvature

In contrast to curves, which do not have intrinsic curvature, but do have extrinsic curvature (they only have a curvature given an embedding), surfaces can have intrinsic curvature, independent of an embedding. The Gaussian curvature, named after Carl Friedrich Gauss, is equal to the product of the principal curvatures, k k. It has a dimension of length and is positive for spheres, negative for one-sheet hyperboloids and zero for planes and cylinders. It determines whether a surface is locally convex (when it is positive) or locally saddle-shaped (when it is negative).

Gaussian curvature is an intrinsic property of the surface, meaning it does not depend on the particular embedding of the surface; intuitively, this means that ants living on the surface could determine the Gaussian curvature. For example, an ant living on a sphere could measure the sum of the interior angles of a triangle and determine that it was greater than 180 degrees, implying that the space it inhabited had positive curvature. On the other hand, an ant living on a cylinder would not detect any such departure from Euclidean geometry; in particular the ant could not detect that the two surfaces have different mean curvatures (see below), which is a purely extrinsic type of curvature.

Formally, Gaussian curvature only depends on the Riemannian metric of the surface. This is Gauss 's celebrated Theorema Egregium, which he found while concerned with geographic surveys and mapmaking.

An intrinsic definition of the Gaussian curvature at a point P is the following: imagine an ant which is tied to P with a short thread of length r. It runs around P while the thread is completely stretched and measures the length C (r) of one complete trip around P. If the surface were flat, the ant would find C (r) = 2π r. On curved surfaces, the formula for C (r) will be different, and the Gaussian curvature K at the point P can be computed by the Bertrand–Diguet–Puiseux theorem as

The integral of the Gaussian curvature over the whole surface is closely related to the surface's Euler characteristic; see the Gauss–Bonnet theorem.

The discrete analog of curvature, corresponding to curvature being concentrated at a point and particularly useful for polyhedra, is the (angular) defect; the analog for the Gauss–Bonnet theorem is Descartes' theorem on total angular defect.

Because (Gaussian) curvature can be defined without reference to an embedding space, it is not necessary that a surface be embedded in a higher-dimensional space in order to be curved. Such an intrinsically curved two-dimensional surface is a simple example of a Riemannian manifold.

### Supplementary Text ###
Surfaces

La courbure des courbes tracées sur une surface est l'outil principal pour définir et étudier la courbure de la surface.

### Draft ###

Query parsing: "fourmi...tour...feuille...courbe" → ant crawling on paper surface, detecting curvature.

Key question: Can intrinsic geometry be detected locally? 

**Answer: Yes, but only Gaussian curvature.**

### 1. Intrinsic vs Extrinsic Curvature

Surface curvature has two components:
- **Gaussian curvature**: K = κ₁κ₂ (product of principal curvatures)
- **Mean curvature**: H = (κ₁+κ₂)/2 (average)

Only Gaussian curvature is "visible" to embedded creatures. Mean curvature requires comparison to embedding space.

### 2. What an Ant Could Measure

🐜 Limited to intrinsic measurements:
- Angles between curves
- Triangle interior angles  
- Circle circumference deviations from 2πr
- Parallel transport phenomena

**Example**: Triangle on sphere → angles sum > 180°
**Example**: Triangle on cylinder → angles sum = 180°

### 3. Detection Mechanisms

Fourmi could use:
- Geometric construction (compass/straightedge equivalent)
- Optical measurements (reflection patterns)
- Mechanical testing (stress, deformation)

**Critical insight**: No ruler, no extrinsic comparison possible.

### 4. Physical Constraints

Paper is 2D manifold → Gaussian curvature = 0 everywhere
∴ Ant would measure Euclidean geometry
Cylinder would also appear "flat" locally

**Exception**: If paper curved globally (sphere-like), ant would detect positive Gaussian curvature.

### 5. Mathematical Formalization

Gaussian curvature K detectable via:
- Angle sum defect in triangles
- Circumference formula deviation: C = 2πr(1 + K·r²/6 + ...)
- Parallel transport of vectors around loops

All require **no embedding space reference**.

⟨H≈0.8⟩ Imagine ant on crumpled paper... geodesic paths following curvature lines... measuring triangle formed by three mountain peaks... sum of angles would reveal intrinsic geometry despite extrinsic distortion... beautiful how mathematics captures this...

**Conclusion**: Fourmi can detect Gaussian curvature through purely intrinsic measurements. Mean curvature, requiring extrinsic comparison, remains invisible.

NB: This is why ants on spheres experience "non-Euclidean" geometry while ants on paper live in Euclidean space.

### Answer###
Oui, une fourmi pourrait effectivement détecter si une surface est courbe, mais seulement en mesurant certaines propriétés géométriques particulières.

La courbure d'une surface peut être détectée intrinsèquement - c'est-à-dire sans référence à l'espace environnant - grâce à des mesures que vous pourriez même réaliser avec un petit insecte. La courbure gaussienne, qui caractérise la façon dont une surface est localement courbée, peut être déterminée par des expériences simples.

Par exemple, si votre fourmi marchait le long d'un cercle tracé sur la feuille de papier et mesurait sa circonférence, elle constaterait que cette circonférence ne suit pas exactement la formule usuelle 2πr (où r est le rayon). Cette déviation de la géométrie euclidienne révèle la courbure de la surface.

De manière plus générale, une fourmi vivant sur une surface sphérique pourrait mesurer les angles intérieurs d'un triangle et constater qu'ils dépassent 180 degrés, lui indiquant que l'espace qu'elle habite a une courbure positive. Inversement, une fourmi sur un cylindre ne détecterait pas de déviation significative de la géométrie euclidienne.

Cette propriété remarquable - le fait que la courbure puisse être détectée intrinsèquement - s'explique par le fait que certaines surfaces, comme les sphères, ont une courbure qui ne dépend pas de la façon dont elles sont placées dans l'espace. C'est une propriété fondamentale qui caractérise ces surfaces de manière intrinsèque.