DefManifold
• DefManifold[manifold, n, {a, b, c, ...}, u] defines manifold to be an n-dimensional differentiable manifold with abstract indices a, b, c,... and ultraindex u on its tangent vbundle.
• DefManifold[manifold, {M1, ..., Mm}, {a, b, c, ...}, u] defines manifold to be the product manifold of previously defined manifolds M1, ..., Mm.
• Currently manifold must be a valid Mathematica symbol, not in use in the present session with any other meaning. In the future a pattern will also be valid to allow for "indexed names".
• The list of all currently defined manifolds is stored in the global variable $Manifolds. All of them have associated upvalues True for the function ManifoldQ. They have type Manifold.
• The dimensionality of the manifold is stored as an upvalue of manifold for the function DimOfManifold. It can be any positive integer or 0. A 0-dim manifold cannot have fields on it. For backwards compatibility, the dimension n can also be given as a list of positive integers, whose length is taken to be n.
• The (nonempty) list of abstract indices {a, b, c, ...} and the ultraindex u are passed to DefVBundle to associate them to the tangent vbundle of the manifold. The name of the tangent vbundle is constructed by default joining the symbol Tangent to the name of the manifold. The result is stored as an upvalue of manifold for the function TangentBundleOfManifold.
• A manifold can have submanifolds (see SubmanifoldsOfManifold) and can be contained in other manifolds, due to the possibility of defining product manifolds. The function SubmanifoldQ checks that a first manifold contains a second manifold.
• Manifolds can have objects (tensors and connections living on the manifold), but in principle they should not be or have servants.
• Options:
Info {"manifold", ""} information on the manifold
Master Null master symbol for the manifold (should be never needed)
PrintAs Identity string or function of symbol for output
ProtectNewSymbol $ProtectNewSymbols whether to protect new symbol or not
• See: Section 4.3.
• See also: UndefManifold, DefVBundle.
• New in version 0.
• Last update: 15-V-2008 for version 0.9.5 of xTensor`.
| Created by Mathematica (May 16, 2008) | |