Twisting By A Line Bundle
An extension of the tensor field idea incorporates an extra line bundle L on M. If W is the tensor product bundle of V with L, then W is a bundle of vector spaces of just the same dimension as V. This allows one to define the concept of tensor density, a 'twisted' type of tensor field. A tensor density is the special case where L is the bundle of densities on a manifold, namely the determinant bundle of the cotangent bundle. (To be strictly accurate, one should also apply the absolute value to the transition functions — this makes little difference for an orientable manifold.) For a more traditional explanation see the tensor density article.
One feature of the bundle of densities (again assuming orientability) L is that Ls is well-defined for real number values of s; this can be read from the transition functions, which take strictly positive real values. This means for example that we can take a half-density, the case where s = ½. In general we can take sections of W, the tensor product of V with Ls, and consider tensor density fields with weight s.
Half-densities are applied in areas such as defining integral operators on manifolds, and geometric quantization.
Read more about this topic: Tensor Field
Famous quotes containing the words twisting, line and/or bundle:
“Farmers in overalls and wide-brimmed straw hats lounge about the store on hot summer days, when the most common sound is the thump-thump-thump of a hound’s leg on the floor as he scratches contentedly. Oldtime hunters say that fleas are a hound’s salvation: his constant twisting and clawing in pursuit of the tormentors keeps his joints supple.”
—Administration in the State of Arka, U.S. public relief program (1935-1943)
“There’s something like a line of gold thread running through a man’s words when he talks to his daughter, and gradually over the years it gets to be long enough for you pick up in your hands and weave into a cloth that feels like love itself. It’s another thing, though, to hold up that cloth for inspection.”
—John Gregory Brown (20th century)
“In the quilts I had found good objects—hospitable, warm, with soft edges yet resistant, with boundaries yet suggesting a continuous safe expanse, a field that could be bundled, a bundle that could be unfurled, portable equipment, light, washable, long-lasting, colorful, versatile, functional and ornamental, private and universal, mine and thine.”
—Radka Donnell-Vogt, U.S. quiltmaker. As quoted in Lives and Works, by Lynn F. Miller and Sally S. Swenson (1981)