Teorema del indice para variedades de contacto
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Una variedad de contacto M es una variedad de dimensión impar 2n+1 equipada con una 1-forma w que no se anula sobre M, y tal quew w^(dw)^n es una forma de volumen. La 1-forma w es llamada una forma de contacto sobre M. Sea H := Ker(w) la distribución diferencial inducida por w . La estructura de contacto w dota al haz vectorial T_HM := H +(TM/H) de una estructura de fibrado de grupos de Heisenberg con la cual T_HM adquiere también una estructura natural de grupoide. De manera análoga a la prueba de Connes [A. Connes, 1984] del teorema de Atiyah-Singer, en [Erp, 2006] se construye el grupoide parabólico tangente T_HM, una variedad cuyo interior es el grupoide M xM x(0,1) y cuya frontera es la unión disyunta de grupoides T_HMU(MxMx1). Como en la prueba del índice de Atiyah-Singer, el grupoide parabólico tangente T_HM es una deformación de grupoides de T_HM en MxM que define un índice topológico ind_t : K0(C*(T_HM)) - Z. Un operador diferencial tipo Rockland P en la variedad de contacto M induce un elemento [σ_H(P)] en K0(C*(T_HM)), el teorema cuya prueba estudiamos en la tesis (ver Teorema 17) afirma que ind_t([σ_H(P)]) = dim(Ker(P)) - dim(Ker(P*)); (0-1) es decir el índice topológico es igual al índice analítico. En el texto tratamos de dar los fundamentos básicos para entender la igualdad (0-1) y su demostración (Texto tomado de la fuente).
A contact manifold M is a 2m + 1 dimensional manifold equipped with a 1-form w such that w does not vanish on M, and w ^ (dw)^n is a volume form. The 1-form w is called a contact form on M. Let H := Ker(w) be the differential distribution induced by w. The contact structure endows to vector bundle T_HM := H x (TM/H) of a structure of principal bundle whose fibers are Heisenberg groups with which THM acquires also a natural groupoid structure. In a similar way to Connes [A. Connes, 1984] proof of Atiyah- Singer theorem, in [Erp, 2006] is builded the parabolic tangent groupoid THM, this is a manifold whose interior is the groupoid M xM x(0, 1) and the boundary is the disjoint union of the groupoids THMU [M xM x1]. As in the Atiyah- Singer index proof, the parabolic tangent groupoid THM is a deformation of groupoids of THM in M x M, this define a topological index indt : K0(C*(T_HM)) -Z. A Rockland type differential operator P in a contact manifold M induces an element [σH(P)] 2 K0(C*(THM)). The theorem that we study in this master thesis (see theorem 17) states that indt([σH(P)]) = dim(Ker(P)) - dim(Ker(P*)); (0-2) that is to say the topological index is the same that the analitical index. We try to provide the basic background to understand the statement and the proof of (0-2).
A contact manifold M is a 2m + 1 dimensional manifold equipped with a 1-form w such that w does not vanish on M, and w ^ (dw)^n is a volume form. The 1-form w is called a contact form on M. Let H := Ker(w) be the differential distribution induced by w. The contact structure endows to vector bundle T_HM := H x (TM/H) of a structure of principal bundle whose fibers are Heisenberg groups with which THM acquires also a natural groupoid structure. In a similar way to Connes [A. Connes, 1984] proof of Atiyah- Singer theorem, in [Erp, 2006] is builded the parabolic tangent groupoid THM, this is a manifold whose interior is the groupoid M xM x(0, 1) and the boundary is the disjoint union of the groupoids THMU [M xM x1]. As in the Atiyah- Singer index proof, the parabolic tangent groupoid THM is a deformation of groupoids of THM in M x M, this define a topological index indt : K0(C*(T_HM)) -Z. A Rockland type differential operator P in a contact manifold M induces an element [σH(P)] 2 K0(C*(THM)). The theorem that we study in this master thesis (see theorem 17) states that indt([σH(P)]) = dim(Ker(P)) - dim(Ker(P*)); (0-2) that is to say the topological index is the same that the analitical index. We try to provide the basic background to understand the statement and the proof of (0-2).