On the existence of Kalb-Ramond wormholes

Static spherically symmetric Kalb-Ramond (KR) traversable wormholes in torsioned spacetime are shown not to exist globally since torsion vanishes not only at the wormhole throat but in all spacetime points which means that all components of the KR field vanish identically. With this result Hochberg-Visser result on the non-violation of the weak energy (WEC) condition in the presence of four-dimensional Einstein-Cartan-Kalb-Ramond (ECKR) geometry with totally skew-symmetric Cartan torsion seems to be a trivial one.