The Flat Case
When M is a Euclidean space and all the fields are taken to be invariant by translations by the vectors of M, we get back to a situation where a tensor field is synonymous with a tensor 'sitting at the origin'. This does no great harm, and is often used in applications. As applied to tensor densities, it does make a difference. The bundle of densities cannot seriously be defined 'at a point'; and therefore a limitation of the contemporary mathematical treatment of tensors is that tensor densities are defined in a roundabout fashion.
Read more about this topic: Tensor Field
Famous quotes containing the words flat and/or case:
“Ask a toad what beauty is, the supreme beauty, the to kalon. He will tell you it is his lady toad with her two big round eyes coming out of her little head, her large flat snout, yellow belly, brown back.”
—Voltaire [François Marie Arouet] (1694–1778)
“I’m here in case you succeed.”
—Dean Devlin, U.S. screenwriter, and Roland Emmerich. Jack O’Neil (Kurt Russell)