Why do we call it "Euclidean Quantum Gravity" instead of "Riemannian Quantum Gravity"? Euclidean quantum gravity is an approach to quantum gravity based on working with Riemannian-signature manifolds and eventually relating the results to our Lorentzian spacetime by means of analytic continuations, for example. I find it curious, at least, that it has the name "Euclidean" rather than "Riemannian", but never bothered much about it until I came across the following comment by Hawking in the abstract of this 1978 paper:

In these lectures I am going to describe an approach to Quantum Gravity using path integrals in the Euclidean regime i.e. over positive definite metrics. (Strictly speaking, Riemannian would be more appropriate but it has the wrong connotations).

What exactly are the wrong connotations suggested by the word "Riemannian" that lead one to avoid it?
 A: The most likely reason I can think of is that Riemann himself, in the 1850s, had suggested the possibility that gravitation might be equivalent to curvature, but his ideas were not successful.  So the use of "Riemannian... gravity" terminology might make people think of the the work associated with Riemann himself—rather than as a distinction with the pseudo-Riemannian geometry of general relativity.
One of the things that special relativity explains is why the coupling constant $q$ is the same for electric and magnetic forces.  Similarly, one of the things that general relativity explains is why the "coupling constant" $m$ is the same for gravitational and fictitious forces.  If gravitation is due to a curvature of the manifold in which bodies are moving, then the commonality of $m$ (that is, the equivalence—or at least proportionality—of the gravitational and inertial masses) is automatic.  Riemann, once he had developed the general theory of curved manifolds, immediately recognized this possibility.  However, since in 1853, Riemann did not know about special relativity, he only considered curvature of 3-dimensional space, instead of (3+1)-dimensional spacetime.
