documentclass{standalone}
usepackage{tikz}
usetikzlibrary{decorations.pathmorphing}

begin{document}
begin{tikzpicture}
node (I)    at ( 4,0)   {I};
node (II)   at (-4,0)   {II};
node (III)  at (0, 2.5) {III};
node (IV)   at (0,-2.5) {IV};

path  % Four corners of left diamond
  (II) +(90:4)  coordinate[label=90:$i^+$]  (IItop)
       +(-90:4) coordinate[label=-90:$i^-$] (IIbot)
       +(0:4)   coordinate                  (IIright)
       +(180:4) coordinate[label=180:$i^0$] (IIleft)
       ;
draw (IIleft) --
          node[midway, above left]    {$cal{J}^+$}
          node[midway, below, sloped] {$bar{u}=infty$}
      (IItop) --
          node[midway, below, sloped] {$bar{u}=0$}
      (IIright) --
          node[midway, below, sloped] {$bar{u}=0$}
      (IIbot) --
          node[midway, above, sloped] {$bar{v}=-infty$}
          node[midway, below left]    {$cal{J}^-$}
      (IIleft) -- cycle;

path % Four conners of the right diamond (no labels this time)
   (I) +(90:4)  coordinate (Itop)
       +(-90:4) coordinate (Ibot)
       +(180:4) coordinate (Ileft)
       +(0:4)   coordinate (Iright)
       ;
% No text this time in the next diagram
draw  (Ileft) -- (Itop) -- (Iright) -- (Ibot) -- (Ileft) -- cycle;

% Squiggly lines
draw[decorate,decoration=zigzag] (IItop) -- (Itop)
      node[midway, above, inner sep=2mm] {$r=0$};

draw[decorate,decoration=zigzag] (IIbot) -- (Ibot)
      node[midway, below, inner sep=2mm] {$r=0$};

end{tikzpicture}
end{document}

usetikzlibrary{decorations.pathmorphing,calc}

draw[->]  % The -> adds an arrow tip
    (Ibot) -- ($(IItop)!.5!(Itop)$);

draw[->]
    (Ibot) to[out=70, in=-15, looseness=1.5] ($(IItop)!.5!(Itop)$);

documentclass{article}
usepackage{tikz}
usetikzlibrary{positioning,decorations.pathmorphing}

begin{document}

begin{tikzpicture}[node distance=2cm]
% coordinates for the nodes
coordinate (A) at (0,0);
coordinate[above right = of A,label=$i^{ast}$] (B);
coordinate[right = 3cm of B,label=$i^{ast}$] (C);
coordinate[below right = of C] (D);
coordinate[below right = of A] (E);
coordinate[right = 3cm of E] (F);
% some straight lines uning the coordinates and adding labels
draw (A) -- node[above] {$f^{ast}$} (B) -- node[below,sloped,pos=0.25] {$bar{v}=0$} node[below,sloped,pos=0.75] {$H^{-}$}
(F) -- (D);
draw (A) -- (E) -- (C) -- (D);
% some decorated lines uning the coordinates and adding labels
draw[decorate,decoration=zigzag] (B) -- node[above] {$r=0$} (C);
draw[decorate,decoration=zigzag] (E) -- node[below] {$r=0$} (F);
end{tikzpicture}

end{document}

documentclass{article}
usepackage{tikz}
usetikzlibrary{decorations.pathmorphing}

begin{document}

begin{tikzpicture}%[scale=2]
pgfmathsetmacromyunit{4}
 draw     (0,0)            node [left] {$i^0$}
       --++(45:myunit)     coordinate (a)
                            node[pos=.5, above left]    {$cal{J}^+$}
                            node[pos=.5, below, sloped] {$bar{u}=infty$}
                            node [above]{$i^+$}
       --++(-45:2*myunit)  node[pos=.25, below, sloped] {$bar{u}=0$}
                            coordinate (d)
                            node [below]{$i^-$}
       --++(45:myunit)     node [right]{$i^0$}
       --++(135:myunit)    coordinate (b)
                            node [above]{$i^+$}
       --++(-135:2*myunit) coordinate (c)
                            node [below]{$i^-$}
       --cycle;

 draw [decorate, decoration=zigzag] (a) -- node[above=6pt] {$r=0$}
                                            node[below=1 cm] {III}
                                            (b)

                                     (c) -- (d);

end{tikzpicture}
end{document}