In differential geometry, we do have a scalar product, which is defined by contracting the indices of two vectors, as in u a v a . If we also had a a tensorial cross product, we would be able to defin...In differential geometry, we do have a scalar product, which is defined by contracting the indices of two vectors, as in u a v a . If we also had a a tensorial cross product, we would be able to define area and volume tensors, so we conclude that there is no tensorial cross product, i.e., an operation that would multiply two rank-1 tensors to produce a rank-1 tensor.
