Scott Walter wrote a 1999 paper on

The non-Euclidean style of Minkowskian relativity, which has a lot of historical info on the use of non-Euclidean geometry for relativity during 1905-1913. After about 1913, the field went fully non-Euclidean to accommodate gravity via general relativity. But before that, there was a split between mathematicians and physics on whether relativity was a non-Euclidean geometry:

Where Minkowski underlined the conceptual continuity of non-Euclidean geometry and the notion of time in relativity, Planck refused the analogy, and emphasized the revolutionary nature of Einstein's new insight.32 For Planck, however, there was at least an historical similarity between non-Euclidean geometry and relativity. The relativity revolution was similar to that engendered by the introduction of non-Euclidean geometry: after a violent struggle, Planck recalled, the Modernisten finally won general acceptance of their doctrine (1910, 42-43).

In his address to the German Association in September, 1910, Planck acknowledged that progress in solving the abstract problems connected with the principle of relativity was largely the work of mathematicians. The advantage of mathematicians, Planck noted (1910, 42), rested in the fact that the "standard mathematical methods" of relativity were "entirely the same as those developed in four-dimensional geometry." Thus for Planck, the space-time formalism had already become the standard for theoretical investigations of the principle of relativity.

Planck's coeditor at the Annalen der Physik , Willy Wien (1864-1928), reiterated the contrast between non-Euclidean geometry and physics in his review of Einstein's and Minkowski's views of space and time. Wien portrayed Einstein's theory of relativity as an induction from results in experimental physics; here, according to Wien (1909, 30), there was "no direct point of contact with non-Euclidean geometry." Minkowski's theory, on the other hand, was associated in Wien's lecture with a different line of development: the abstract, speculative theories of geometry invented by mathematicians from Carl Friedrich Gauss to David Hilbert.

Wien admitted there was something "extraordinarily compelling" about Minkowski's view. The whole Minkowskian system, he said, "evokes the conviction that the facts would have to join it as a fully internal consequence." As an example of this, he mentioned Minkowski's four equations of motion, the fourth of which is also the law of energy conservation (see Minkowski, 1909, p. 85). Wien nonetheless distanced himself from the formal principles embodied in Minkowski's contribution to relativity when he recalled that the physicist's credo was not aesthetics but experiment. "For the physicist," Wien concluded, "Nature alone must make the final decision."33 ...

In summary, certain mathematicians and physicists cast Minkowski's work in a tradition of research on non-Euclidean geometry. For the mathematicians Mansion and Mathews, relativity theory was ripe for study and development by geometers. The physicists Planck and Wien, on the other hand, denied any link between non-Euclidean geometry and Einstein's theory of relativity. ...

Wien, for one, had silently retracted his opinion (see § 4 above), by excising the offending passage of his 1909 lecture for reedition ...

It is commonly remarked that special relativity was quickly accepted, even tho it was a radical shift from Newtonian mechanics. But this shows that only the mathematicians accepted the non-Euclidean geometrical core of the theory, while the physicists Einstein, Planck, and Wien did not.

Here is a striking example:

In a footnote to this work, Sommerfeld remarked that the geometrical relations he presented in terms of three real and one imaginary coordinate could be reinterpreted in terms of non-Euclidean geometry. The latter approach, Sommerfeld cautioned (1910a, 752), could "hardly be recommended."

Equally omitted from Varicak's report was his explanation of the Lorentz-FitzGerald contraction (according to which all moving bodies shrink in their direction of motion with respect to the ether) as a psychological phenomenon. Earlier in the year, Einstein had contested his argument by maintaining the reality of the contraction. 46

Thus ignoring both Sommerfeld's dim view of his non-Euclidean program, and Einstein's correction of his interpretation of relativity theory, Varicak went on to demonstrate the formal simplicity afforded by hyperbolic functions in the theory of relativity.

Calling the contraction a "psychological phenomenon" sounds bogus, but Varicak's point was that the contraction is just a non-Euclidean geometrical artifact, and objects just seem contracted because of the way we use Euclidean coordinates.

Varicak was completely correct about this. So were Mathematicians Poincare and Minkowski at the time, and that is how modern textbooks describe the contraction.

Walter refers to "Einstein's correction of his interpretation". Varicak thought that he was agreeing with Einstein, but Einstein actually published a paper in 1911 specifically disavowing Varicak's geometrical interpretation, and insisting on Lorentz's view of objects actually contracting by motion thru the aether or whatever. For details, see

The Einstein-Varicak Correspondence on Relativistic Rigid Rotation.

Varicak's paper started:

The occurrence of Ehrenfest's paradox is understandable, when one clings to the standpoint taken by Lorentz in the formulation of his contraction hypothesis, i.e., when one sees the contraction of moving rigid bodies in the direction of motion as a change which takes place in an objective way. Every element of the periphery will be changed independently of the observer according to Lorentz, while the elements of the radius remain non-contracted.

However, if one employs Einstein's standpoint, according to whom the mentioned contraction is only an apparent, subjective phenomenon, caused by the manner of our clock-regulation and length-measurement, then this contradiction doesn't appear to be justified.

Varicak is completely correct, except that he is attributing the Poincare-Minkowski geometric view of relativity to Einstein. Einstein published a rebuttal to this, where he sided with Lorentz's interpretation of the contraction.

For his source, Varicak cited a 1909 paper. Two Americans wrote

The Principle of Relativity, and Non-Newtonian Mechanics:

Let us emphasize once more, that these changes in the units of time and length, as well as the changes in the units of mass, force, and energy which we are about to discuss, possess in a certain sense a purely factitious significance; although, as we shall show, this is equally true of other universally accepted physical conceptions. We are only justified in speaking of a body in motion when we have in mind some definite though arbitrarily chosen point as a point of rest. The distortion of a moving body is not a physical change in the body itself, but is a scientific fiction.

When Lorentz first advanced the idea that an electron, or in fact any moving body, is shortened in the line of its motion, he pictured a real distortion of the body in consequence of a real motion through a stationary ether, and his theory has aroused considerable discussion as to the nature of the forces which would be necessary to produce such a deformation. The point of view first advanced by Einstein, which we have here adopted, is radically different. Absolute motion has no significance. Imagine an electron and a number of observers moving in different directions with respect to it. To each observer, naÃ¯vely considering himself to be at rest, the electron will appear shortened in a different direction and by a different amount; but the physical condition of the electron obviously does not depend upon the state of mind of the observers.

Although these changes in the units of space and time appear in a certain sense psychological, we adopt them rather than abandon completely the fundamental conceptions of space, time, and velocity, upon which the science of physics now rests. At present there appears no other alternative.

The paper endorses views that "appear in a certain sense psychological". By that, he means that our measures of distance and time require referring to an observer. That is the geometric relativistic view that nearly everyone accepts today.

You would think that Einstein would jump at the chance to claim credit for having a superior understanding than what Lorentz had, and to accept credit for the more modern geometrical interpretation. By 1911, the geometrical interpretation was widely understood and accepted. He must have strongly disagreed with the geometrical interpretation.

I don't see how anyone can credit Einstein for special relativity in view of the fact that he disavowed the geometrical interpretation.

The non-Euclidean geometry of relativity was discovered by Poincare in 1905, popularized by Minkowski in 1907, and widely accepted in 1908. In 1911, Einstein still did not understand or accept it.

This is a reason I do not credit Einstein for relativity. It is not that he and Poincare published the same thing in 1905. It is that Poincare was 6 years ahead of Einstein.

Walter's historical analysis ends in about 1913, so he gives the impression that the use of non-Euclidean geometry in relativity was just some topic to amuse mathematicians, and was a big dead-end for physicists. In fact, the non-Euclidean geometry view of relativity has dominated from 1913 to the present day.