From the viewpoint of modern geometry, is there a "best" definition of the term "triangle"?

Question

I can think of at least six different possible definitions of the term "triangle" in Euclidean geometry.

1. a subset of $\mathbb{R}^n$ that can be expressed as the convex hull of three or fewer points.
2. a subset of $\mathbb{R}^n$ that can be expressed as the convex hull of three points, but no fewer.
3. A function $\{0,1,2\} \rightarrow \mathbb{R}^n$.
4. A function $\{0,1,2\} \rightarrow \mathbb{R}^n$ whose multiset image is an affinely independent subset of $\mathbb{R}^n$.
5. A point $p \in \mathbb{R}^n$ together with two vectors in $T_p,$ the tangent space at $p$.
6. A point $p \in \mathbb{R}^n$ together with two linearly independent vectors in $T_p$.

From the viewpoint of modern geometry, is there a "best" definition of the term "triangle" that generalizes most easily to e.g. Riemannian manifolds and beyond? And what framework (e.g. Riemannian manifolds) does this definition belong to?

Answer 1

The most common notion of a "triangle" used in the modern metric and Riemannian geometry is "none of the above". The relevant notion is the one of a "geodesic triangle".

Definition 1. Suppose that $(X,d)$ is a metric space. A geodesic triangle in $X$ is a concatenation of three geodesic segments $\gamma_1, \gamma_2, \gamma_3$ where $\gamma_i$ connects $x_i, x_{i+1}$, $i$ is taken modulo 3.

In this definition, geodesic means an isometric embedding of an interval in the real line into $X$.

Definition 2. Suppose that $(M,g)$ is a Riemannian manifold. A geodesic triangle in $X$ is a concatenation of three geodesic segments $\gamma_1, \gamma_2, \gamma_3$ where $\gamma_i$ connects $x_i, x_{i+1}$, $i$ is taken modulo 3.

In this definition, geodesic means a smooth path $\gamma$ of zero acceleration: $\gamma''=0$, where the second derivative is taken via the Levi-Civita connection on $(M,g)$.

These notions of triangles provide a key tool to:

(1) Defining upper and lower curvature bounds in metric spaces which are not Riemannian manifolds.

(2) Connecting upper and lower curvature bounds for Riemannian metrics to the global geometric and topological properties of the manifolds. ("Comparison Geometry".)

(3) Introducing geometric ideas into the combinatorial group theory.

Read more in:

1. Cheeger and Ebin, "Comparison Theorems in Riemannian Geometry".

2. Burago, Burago, Ivanov, "A Course in Metric Geometry".

3. Bridson and Haefliger, "Metric Spaces of Non-Positive Curvature".