where is an -dimensional bounded rectifiable set, is an orientation of i.e. is a unit -vector representing the approximate tangent plane of at for -a.e. and, finally, is an integer valued measurable function defined a.e. on (called multiplicity). The space of -dimensional rectifiable currents is denoted by .
An -dimensional rectifiable current such that the boundary is itself an -dimensional rectifiable current, is called integral current. The space of integral currents is denoted by . We point out that the word “integral” refers to the fact that the multiplicity is integer valued.
Also notice that rectifiable and integral currents are not vector subspaces of the space of currents. In fact while the sum of two rectifiable currents is again a rectifiable current, the multiplication by a real number gives a rectifiable current only if the number is an integer.
The compactness theorem makes the space of integral currents a good space where geometric problems can be ambiented.
On rectifiable currents one can define an integral flat norm
The closure of the space under the integral flat norm is called the space of integral flat chains and is denoted by .
As a consequence of the closure theorem, one finds that where is the mass norm of a current.
|Date of creation||2013-03-22 14:28:34|
|Last modified on||2013-03-22 14:28:34|
|Last modified by||paolini (1187)|
|Defines||integral flat norm|
|Defines||integral flat chains|