In paper : https://arxiv.org/pdf/1908.00227.pdf it is stated that
In the metric TSP problem, which we study here, the distances satisfy the triangle inequality. Therefore the problem is equivalent to finding a closed Eulerian connected walk of minimum cost.
They don't seem to be equivalent at all, since for a complete graph of size 5, with all edge costs 1, a metric TSP solution would be of cost (and length) 5, whereas an Eulerian cycle would be of cost (and length) 10. Am I misunderstanding?