I was studying the mathematical description of gauge theories (in terms of bundle, connection, curvature,…) and something bothers me in the terminology when I compare it with general relativity.
In gauge theory, take a vector bundle $E \rightarrow M$ over a Riemannian Manifold $(M,g)$ and suppose you have a connection $\nabla: \Gamma(E) \longrightarrow \Gamma(E) \otimes \Gamma(T^*M)$. Then, you defined the curvature tensor $F_\nabla$ associated to your connection $\nabla$ as an $End(E)$-valued 2 form i.e $F_\nabla \in \Omega^2(M,End(E))$
$$ \nabla \circ \nabla := F_\nabla$$
where you extend $\nabla$ on the $p$-form.
Equivalently you can defined the curvature tensor as a map $F_\nabla : TM \times TM \times E \rightarrow E$ such that
$$ F_\nabla(X,Y)s=\nabla_X\nabla_Y s – \nabla_Y\nabla_X s – \nabla_{[X,Y]}s \qquad X,Y \in \Gamma(TM), s \in \Gamma(E)$$
Then, $F_\nabla$ give you information on the curvature of your bundle i.e if you take a vector $v \in E$ and you do the parallel transport of $v$ on a closed path with respect to the connection $\nabla$, the difference between $v$ and its parallel transport $v'$ will be given by $F_\nabla$.
Then, you remark that if as vector bundle you take $E=TM$ and as connection you take the Levi-Civita connection $\nabla=\nabla^{LC}$ you see that your curvature tensor associated to $\nabla$ is exactly the Riemann curvature tensor i.e
$$ F_\nabla = R$$
Now my question:
In the literature, the Riemann curvature tensor is often called the curvature of space-time. But in view of the definition of the curvature tensor on a vector bundle, why is $R$ not rather the curvature tensor of the tangent bundle (associated with the Levi-civita connection)?
(My answer would be: I suppose that in fact the Riemann tensor corresponds to the curvature tensor of tangent bundle (associated with the Levi-Civita connection) but it also gives plenty (if not all) of information on the curvature of space-time (since tangent space is very linked to the manifold) and therefore it is called curvature of space-time (and therefore it would be a problem of terminology). I'm not sure of this answer that's why I wanted an outside opinion.)
Best Answer
The answer is that when we talk about the intrinsic curvature of a manifold, what we're talking about is the curvature of its tangent bundle. That's what we mean when we talk about spacetime being curved.
In other words, intrinsic curvature is characterized by how tangent spaces to different points on the manifold are connected via parallel transport. If the mapping from the tangent space at $p$ to the tangent space at $q$ is path-independent, then the space is said to be (intrinsically) flat. Otherwise, it possesses non-zero intrinsic curvature which is quantified by the Riemann tensor.