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The function ", StyleBox["hopf", FontFamily->"Arial", FontSlant->"Italic"], StyleBox["[\[Theta], \[Phi], \[Alpha]] ", FontFamily->"Arial"], "returns the stereographic projection of a point from the fibre over (\ \[Theta], \[Phi]). The parameter \[Alpha] describes a particular point in the \ fibre." }], "Text", CellMargins->{{Inherited, 139}, {Inherited, Inherited}}, TextJustification->1, FontFamily->"Arial", FontSize->14], Cell[BoxData[ \(\(\(hopf[\[Theta]_, \ \[Phi]_, \ \[Alpha]_] := ster[Cos[\[Theta]/2] Cos[\[Alpha]], Cos[\[Theta]/2] Sin[\[Alpha]], \ Sin[\[Theta]/2] Cos[\ \[Alpha] + \ \[Phi]], \ Sin[\[Theta]/2] Sin[\ \[Alpha] + \ \[Phi]]\ ];\)\(\ \ \ \ \)\)\)], "Input", CellLabel->"In[31]:=", CellMargins->{{Inherited, 139}, {Inherited, Inherited}}], Cell[TextData[{ "The function ", StyleBox["fibra ", FontWeight->"Bold", FontSlant->"Italic"], "draws the fiber over a point ", "(\[Theta], \[Phi])", " under the stereographics projection. The fibers project into circles, and \ only those passing through the north pole of the sphere ", Cell[BoxData[ \(TraditionalForm\`S\^3\)]], " project into the line. Note that the color and thickness of the projected \ fibre can be given." }], "Text", FontFamily->"Arial", FontSize->14], Cell[BoxData[ StyleBox[\(fibra[\[Theta]_, \[Phi]_, \ \ prop_: {RGBColor[1, 0, 0], \ Thickness[0.001]}]\ := \ ParametricPlot3D[ Append[hopf[\[Theta], \ \ \[Phi], \ t], prop], \ {t, 0, 2 \[Pi]}, \ Axes \[Rule] False, \ Boxed \[Rule] False, \ DisplayFunction \[Rule] Identity];\), FontSize->14]], "Input", CellLabel->"In[32]:=", FontFamily->"Arial"], Cell[BoxData[ \(Now, \ we\ choose\ an\ arbitrary\ orthonormal\ basis\ \((e1, \ e2, \ e3)\)\ of\ R\^3 . \ We\ need\ it\ in\ order\ to\ parameterize\ circles\ of\ the\ sphere\ \ S\^2\ laying\ in\ the\ planes\ normal\ to\ e1 . \ You\ can\ change\ the\ basis\ as\ you\ like . \ What\ you\ get\ is\ geometrically\ the\ same, \ \ \(\(anyway\)\(.\)\)\)], "Text", TextAlignment->Left, TextJustification->1, FontFamily->"Arial", FontSize->14], Cell[BoxData[{ \(\(e1\ = \ 1/Sqrt[2] {1, \ \(-1\), \ 0} // N;\)\), "\[IndentingNewLine]", \(\(e2\ = \ N[1/Sqrt[3] {1, \ 1, \ 1}];\)\), "\[IndentingNewLine]", \(\(e3\ = \ 1/Sqrt[6] {1, \ 1, \ \(-2\)} // N;\)\)}], "Input", CellLabel->"In[6]:="], Cell[TextData[{ "The function ", StyleBox["krug ", FontWeight->"Bold", FontSlant->"Italic"], " parameterizes the circle of radius ", "Sqrt[1-r^2]", " with the parameter ", "\[Phi]. The circle lies in the plane orthogonal to e1. The functions ", StyleBox["th", FontWeight->"Bold", FontSlant->"Italic"], " and ", StyleBox["ph ", FontWeight->"Bold", FontSlant->"Italic"], " are auxilary functions for determining ", "(\[Theta], \[Phi])", " angles for point (x, y, z) from the sphere ", Cell[BoxData[ \(TraditionalForm\`S\^2\)]], "." }], "Text", FontFamily->"Arial", FontSize->14], Cell[BoxData[{ \(\(krug[\[Phi]_, \ r_] := N[r\ e1\ + \ Sqrt[1 - r^2] \((Cos[\[Phi]] e2 + \ Sin[\[Phi]] e3)\)];\)\), "\[IndentingNewLine]", \(\(th[{x_, \ y_, \ z_}] := \ N[ArcCos[z]];\)\), "\[IndentingNewLine]", \(\(ph[{x_, \ y_, \ z_}] := \ N[ArcTan[x, \ y]];\)\)}], "Input", CellLabel->"In[9]:="], Cell[TextData[{ "The function fibreKruga[r] all the fibres (i.e. torus) over the circle (of \ radius Sqrt[1-r^2]) from ", Cell[BoxData[ \(TraditionalForm\`S\^2\)]], ". You wan't see anything since ", "DisplayFunction\[Rule]Identity is set." }], "Text", FontFamily->"Arial", FontSize->14], Cell[BoxData[ \(\(fibreKruga[r_, \ prop_: {RGBColor[0, 0, 1], \ Thickness[0.001]}] := \ Table[ParametricPlot3D[ Append[hopf[th[krug[j, r]], ph[krug[j, r]], \ t], \ prop], \ {t, 0, 2 \[Pi]}, \ Axes \[Rule] False, \ Boxed \[Rule] False, \ DisplayFunction \[Rule] Identity], \ \ {j, 0, 2\ Pi, \ 2 Pi/20}];\)\)], "Input", CellLabel->"In[26]:="], Cell[TextData[{ "Here comes the first animation of the Hopf bundle. The twisted torus in \ each moment is a stereographic projection of all fibers over a single circle \ from ", Cell[BoxData[ \(TraditionalForm\`\(\(S\^2\)\(.\)\)\)]], " In the extreme cases r = t = \[PlusMinus] 1 the circle is point and there \ is only one fiber (circle) above it. The tori fill the whole ", Cell[BoxData[ \(TraditionalForm\`R\^3\)]], " taking each point only once." }], "Text", FontFamily->"Arial", FontSize->14], Cell[BoxData[ \(\(Table[ Show[fibreKruga[t, \ {Hue[t/2 - 1/2], \ Thickness[0.005]}], \ DisplayFunction \[Rule] $DisplayFunction, \ ViewPoint -> {1.011, \ 0.179, \ 3.224}, \ PlotRange \[Rule] 3.9 {{\(-1\), 1}, \ {\(-1\), 1}, \ {\(-1\), 1}}], \ {t, \(-1\), 1, \ 0.1}];\)\)], "Input", CellLabel->"In[13]:="], Cell[TextData[{ "The fibers over two congruent circles of the ", Cell[BoxData[ \(TraditionalForm\`S\^2\)]], " (obtained for r = \[PlusMinus]0.8) are linked tori." }], "Text", FontFamily->"Arial", FontSize->14], Cell[BoxData[ \(\(Show[ fibreKruga[\(-0.8\), \ {RGBColor[0, 0, 1], \ Thickness[0.001]}], \ \[IndentingNewLine]fibreKruga[ 0.8, \ {RGBColor[1, 0, 0], \ Thickness[0.001]}], \ DisplayFunction \[Rule] $DisplayFunction, \ ViewPoint -> {1.011, \ 0.179, \ 3.224}];\)\)], "Input", CellLabel->"In[27]:="], Cell[TextData[{ "This is well known animation of the Hopf bundle where circles are normal \ to the z-axis (the same one would get by setting e1={0,0,1}, e2={0,1,0}, \ e3={1,0,0} in the previos method). Note that the projection of the last fiber \ is a line. That is because that fiber (over the point (0,0,1) of ", Cell[BoxData[ \(TraditionalForm\`S\^2\)]], ") and only it, passes through the north pole of ", Cell[BoxData[ \(TraditionalForm\`S\^3\)]], "." }], "Text", FontFamily->"Arial", FontSize->14], Cell[BoxData[ \(\(Table[ Show[Table[ fibra[t, phi, \ {RGBColor[t/Pi, \ phi/\((2 Pi)\), t/Pi], \ Thickness[0.001]}], \ {phi, \ 0, \ 2\ Pi, \ 2\ Pi/20}], \ DisplayFunction \[Rule] $DisplayFunction, PlotRange \[Rule] 1.5 {{\(-1\), 1}, \ {\(-1\), 1}, \ {\(-1\), 1}}], \ {t, \ 0\ , \ Pi, \ Pi/16}];\)\)], "Input", CellLabel->"In[33]:="], Cell["Here is a funny way to make a torus out of circles. ", "Text", FontFamily->"Arial", FontSize->14], Cell[BoxData[{ \(\(slika\ = \ {};\)\), "\[IndentingNewLine]", \(\(Table[ Show[slika\ = \ Append[slika, \ fibra[Pi/4, \ t]], \ DisplayFunction \[Rule] $DisplayFunction, \ PlotRange \[Rule] 1.5 {{\(-1\), 1}, \ {\(-1\), 1}, \ {\(-1\), 1}}], \ {t, 0, 2\ Pi, \ 2\ Pi/20}];\)\)}], "Input", CellLabel->"In[34]:="], Cell["\<\ To get the whole bundle we have to glue two its trivializations. Now, we show \ that over the equator this gluing is realized as a twist. The black torus is \ a bundle over the equator in one of the trivializations. The red one is the \ same part of the bundle but parameterized in the other trivializations. The \ way of gluing is governed by the transition function, and here is seen as a \ twist.\ \>", "Text", FontFamily->"Arial", FontSize->14], Cell[BoxData[{ \(\(torus\ = \ Table[ParametricPlot3D[ hopf[Pi/4, \ phi, \ \ \ a0], \ \ {phi, \ 0, \ 2\ Pi}, Boxed \[Rule] False, \ Axes \[Rule] False, \ DisplayFunction \[Rule] Identity], \ {a0, 0\ , 2\ Pi\ , \ \ 2 Pi/10}];\)\), "\[IndentingNewLine]", \(\(Table[ Show[Append[ Table[\[IndentingNewLine]ParametricPlot3D[\[IndentingNewLine]\ Append[hopf[Pi/4, \ phi, \ \ \ phi\ + \ a0], {RGBColor[1, 0, 0], \ Thickness[0.001]}\ ], \ {phi, \ \(-a0\), \ t - \ a0}, \ Boxed \[Rule] False, \ Axes \[Rule] False, \ DisplayFunction \[Rule] Identity], \ \[IndentingNewLine]{a0, 0\ , 2\ Pi\ , \ \ 2 Pi/10}], \ torus], \ DisplayFunction \[Rule] $DisplayFunction, \ PlotRange \[Rule] 1.5 {{\(-1\), 1}, \ {\(-1\), 1}, \ {\(-1\), 1}}], \ {t, 2 Pi/20, \ 2\ Pi, \ 2\ Pi\ /20}];\)\)}], "Input", CellLabel->"In[36]:="] }, Open ]], Cell[CellGroupData[{ Cell["The Dirac Connection on the Hopf Bundle", "Subtitle"], Cell["\<\ Now, I will go into more details in order to estabilish the notation. Let \[Pi]: P \[RightArrow] M is a principal bundle with the base space M, the \ total space P, the projection map \[Pi] and with the structure group G. We \ also denote it by G\[RightArrow]P\[RightArrow]M. Let g be the Lie algebra of \ the group G.\ \>", "Text", TextAlignment->Left, TextJustification->1, FontFamily->"Arial", FontSize->14], Cell[BoxData[ \(According\ to\ one\ possible\ definition, \ a\ connection\ on\ a\ principal\ bundle\ \ G \[RightArrow] P \[RightArrow] M\ \ is\ a\ globally\ defined\ form\ \(\[Omega] : P \[RightArrow] \(T\^*\) P\[CircleTimes]g\), \ on\ P\ with\ values\ in\ the\ Lie\ algebra\ g . \ The\ form\ \[Omega]\ is\ also\ required\ to\ be\ invariant\ under\ \ the\ natural\ action\ of\ the\ group\ G\ on\ \(\(P\)\(.\)\)\)], "Text", FontFamily->"Arial", FontSize->14], Cell[BoxData[ \(If\ \(\[Alpha] : \([a, \ b]\) \[RightArrow] M\ is\ a\ \ curve\ \ on\ a\ base\ space\), \ \ \ curve\ \ \(\(\[Alpha]\&_\) : \([a, \ b]\) \[RightArrow] P\ is\ a\ lift\ of\ the\ curve\ \[Alpha]\ provided\ \[Pi]\ \[SmallCircle]\ \[Alpha]\&_\)\ = \ \[Alpha] . \ Obviously, \ the\ lift\ of\ \[Alpha]\ is\ not\ unique . \ However, \ given\ a\ connection\ \[Omega], \ we\ can\ define\ the\ horizontal\ lift\ \ \[Alpha]\&_\ of\ \[Alpha]\ with\ \ additional\ condition\ \[Omega] \((\ \[Alpha]\&_')\)\ = \ \(0\ in\ each\ \ point\ of\ \[Alpha]\&_\ . \ \ If\ the\ initial\ condition\ \ \(\[Alpha]\&_\) \ \((a)\)\ = \ p\ \[Element] \ P\), \ is\ given\ \((p \[Element] \ \(\[Pi]\^\(-1\)\) \((\[Alpha] \((a)\))\))\), \ \ the\ horizontal\ lift\ of\ a\ curve\ \[Alpha]\ is\ unique . \ All\ horizontal\ lifts\ \((with\ different\ initial\ conditions)\)\ \ of\ the\ curve\ \[Alpha]\ are\ parameterized\ by\ G, \ i . e . \ G\ acts\ transitively\ on\ P\ transforming\ one\ horizontal\ lift\ \ into\ \(\(another\)\(.\)\)\)], "Text", FontFamily->"Arial", FontSize->14], Cell[BoxData[ \(The\ Hopf\ fibration\ is\ a\ principal\ bundle\ with\ a\ base\ space\ M\ \ = \ S\^2, \ total\ space\ P\ = \ \(S\^3\ and\ the\ structure\ group\ G\ = \(U \((1)\ \)\ = \ S\^1 . \ We\ consider\ the\ total\ space\ S\^3\ embedded\ in\ the\ \ complex\ plane\)\), \ i . e . \ S\^3 \[Subset] R\^4 \[Subset] \ C\^2, \ with\ coordinates\ z\ = \ \(\((z\_1, \ z\_2)\)\(.\)\)\)], "Text", FontFamily->"Arial", FontSize->14], Cell[BoxData[ \(One\ can\ check\ that\ \ well\ known\ Dirak\ connection\ can\ be\ \ written\ in\ the\ form\ \[Omega]\ = \ \(z\ d z\&_\ = \ \(z\_1\) \(dz\_1\)\&_ + \ \(z\_\(\(2\)\(\ \)\)\) \ \(dz\_2\)\&_ . \ This\ is\ an\ example\ of\ \ connection\ on\ the\ Hopf\ \ \(\(bundle\)\(.\)\)\)\)], "Text", FontFamily->"Arial", FontSize->14], Cell[TextData[{ "Now, we are going to visualize the Dirak connection via horizontal lift. \ Let \[Alpha][t] = sfera[phi[t], theta[t]], t \[Element]", " [a, b], ", "be an arbitrary curve on the sphere ", Cell[BoxData[ \(TraditionalForm\`S\^2\)]], ". For instance:" }], "Text", FontFamily->"Arial", FontSize->14], Cell[BoxData[{ \(a\ = 0; \ b = Pi;\), "\[IndentingNewLine]", \(\(phi[t_] := 1/2 Sin[2 t];\)\), "\[IndentingNewLine]", \(\(theta[t_] := t;\)\)}], "Input", CellLabel->"In[38]:="], Cell["The standard parameterization of the sphere.", "Text", FontFamily->"Arial", FontSize->14], Cell[BoxData[ \(sphere[theta_, \ phi_] := \ {Cos[theta]\ Cos[phi], \ Cos[theta] Sin[phi], \ Sin[theta]}\)], "Input", CellLabel->"In[41]:="], Cell["We can see how the original curve looks on the sphere.", "Text", FontFamily->"Arial", FontSize->14], Cell[BoxData[{ \(\(sphCurve\ = \ ParametricPlot3D[ Append[sphere[phi[t], \ theta[t]], {Thickness[0.01]}], \ {t, \ a, \ b}, Boxed \[Rule] False, \ Axes \[Rule] False, \ DisplayFunction \[Rule] Identity];\)\), "\n", \(\(sph\ = \ ParametricPlot3D[ Append[sphere[\[Theta], \ \[Phi]], \ {\ EdgeForm[]}], \ {\[Theta], \ 0, \ 2\ Pi, \ 2 Pi/40}, \ {\[Phi], \ \(-Pi\)/2, \ Pi/2}, \ Boxed \[Rule] False, \ Axes \[Rule] False, \ DisplayFunction \[Rule] Identity];\)\), "\n", \(\(Show[sph, \ sphCurve, \ DisplayFunction \[Rule] $DisplayFunction, ViewPoint -> {2.240, \ 2.536, \ 0.042}];\)\)}], "Input", CellLabel->"In[45]:="], Cell[BoxData[ RowBox[{\(We\ are\ going\ to\ draw\), ",", " ", RowBox[{ "the", " ", "fibers", " ", "over", " ", "the", " ", "points", " ", "of", " ", "the", " ", \(curve . \ The\), " ", StyleBox["start", FontWeight->"Bold", FontSlant->"Italic"], " ", "and", " ", StyleBox["end", FontWeight->"Bold", FontSlant->"Italic"]}], StyleBox[",", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[" ", FontWeight->"Plain", FontSlant->"Italic"], StyleBox[\(are\ fibers\ over\ the\ starting\ and\ ending\ points\ of\ \ the\ curve\ \((they\ are\ blue\ and\ green\ coloured)\) . \ The\ lift\ of\ the\ curve\ is\ going\ to\ start\ and\ finish\ at\ \ those\ \(\(fibers\)\(.\)\)\), FontSlant->"Italic"]}]], "Input", TextAlignment->Left, TextJustification->1, FontFamily->"Arial", FontSize->14, FontWeight->"Plain"], Cell[BoxData[{ \(\(fibers\ = \ Show[Table[ fibra[theta[t], \ phi[t]], \ {t, \ a\ , \ \ b, \ \ Abs[a - b]/20}], \ DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(start\ := \ fibra[theta[a], \ phi[a], \ {RGBColor[0, 1, 0], \ Thickness[0.008]}];\)\), "\[IndentingNewLine]", \(\(end := \ fibra[theta[b], \ phi[b], \ {RGBColor[0, 0, 1], \ Thickness[0.008]}];\)\), "\[IndentingNewLine]", \(\(Show[fibers, \ start, \ end, \ DisplayFunction \[Rule] $DisplayFunction];\)\)}], "Input", CellLabel->"In[48]:=", TextAlignment->Left, TextJustification->1], Cell[BoxData[ RowBox[{ "One", " ", "can", " ", "calculate", " ", "the", " ", "differential", " ", "equation", " ", "of", " ", "the", " ", "lift", " ", "which", " ", "depends", " ", "on", " ", "the", " ", "initial", " ", \(condition . The\), " ", "function", " ", StyleBox["solution", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[ RowBox[{ StyleBox[" ", FontWeight->"Bold", FontSlant->"Italic"], StyleBox[" ", FontSlant->"Italic"]}]], "is", " ", "a", " ", "numerical", " ", "solution", " ", "of", " ", "that", " ", "differential", " ", "equation", " ", "and", " ", "the", " ", "function", " ", StyleBox["lift", FontWeight->"Bold"], " ", "draws", " ", "the", " ", "stereographic", " ", "projection", " ", "of", " ", "the", " ", "lift", " ", \(\((depending\ on\ the\ initial\ condition)\)\(.\)\)}]], "Input", FontFamily->"Arial", FontSize->14, FontWeight->"Plain"], Cell[BoxData[{ \(\(solution[m_]\ := \ NDSolve[{\(\[Alpha]'\)[ u] \[Equal] \(-\ Sin[theta[u]/2]^2\)\ \(phi'\)[u], \ \[Alpha][ 0]\ \[Equal] Evaluate[m]}\ , \[Alpha], \ {u, \ 0, \ 2\ Pi}];\)\), "\n", \(\(lift[m_]\ := \ ParametricPlot3D[ Append[hopf[theta[t], \ phi[t], Evaluate[\[Alpha][t] /. \(solution[m]\)[\([1]\)]]], \ {RGBColor[ 0, 0, 0], \ Thickness[0.008]}], \ {t, \ a, \ \ b, \ Abs[a - b]/40}, \ Boxed \[Rule] False, \ Axes \[Rule] False, \ DisplayFunction \[Rule] Identity];\)\)}], "Input", CellLabel->"In[52]:="], Cell[BoxData[ \(Here\ is\ the\ animation\ of\ all\ the\ lifts\ depending\ on\ the\ \ initial\ condition . One\ can\ think\ of\ this\ animation\ as\ of\ an\ action\ of\ a\ \ group\ U \((1)\)\ on\ a\ piece\ of\ Hopf\ bundle . \ The\ action\ transformes\ one\ horizontal\ lift\ into\ another\ lift\ \ of\ the\ same\ \(\(curve\)\(.\)\)\)], "Input", TextAlignment->Left, TextJustification->1, FontFamily->"Arial", FontSize->14, FontWeight->"Plain"], Cell[BoxData[ \(\(Table[ Show[fibers, \ lift[m], start, end, DisplayFunction \[Rule] $DisplayFunction, \ SphericalRegion \[Rule] True\ , \ PlotRange \[Rule] 1.8 {{\(-1\), 1}, \ {\(-1\), 1}, \ {\(-1.5\), 1.5}}], \ {m, \ 0, \ 2\ Pi, \ 2\ Pi/20}];\)\)], "Input", CellLabel->"In[54]:="], Cell["\<\ Now we take another example: the curve is the \[Theta] = \[Pi]/4 parallel \ (the one passing through the Belgrade) of the sphere. \ \>", "Text", FontFamily->"Arial", FontSize->14], Cell[BoxData[{ \(a\ = 0; \ b = 2\ Pi;\), "\[IndentingNewLine]", \(\(theta[t_] := Pi/4;\)\), "\[IndentingNewLine]", \(\(phi[t_] := t;\)\)}], "Input", CellLabel->"In[93]:="], Cell[BoxData[ RowBox[{ StyleBox["Since", FontWeight->"Plain"], StyleBox[",", FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"], RowBox[{ StyleBox["it", FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"], StyleBox["is", FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"], StyleBox["a", FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"], StyleBox["closed", FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"], StyleBox["curve", FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"], StyleBox["instead", FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"], StyleBox["the", FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"], StyleBox["start", FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"], StyleBox["and", FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"], StyleBox["end", FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"], StyleBox["fibre", FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"], StyleBox["we", FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"], StyleBox["pick", FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"], StyleBox["only", FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"], StyleBox["one", FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"], StyleBox["fibre", FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"], StyleBox["named", FontWeight->"Plain"], StyleBox[" ", FontWeight->"Plain"], RowBox[{ StyleBox["theFiber", FontSlant->"Italic"], "."}]}]}]], "Input", TextAlignment->Left, TextJustification->1, FontFamily->"Arial", FontSize->14], Cell[BoxData[{ \(\(torus\ = \ Show[Table[ fibra[theta[t], \ phi[t]], \ {t, \ a\ , \ \ b, \ \ Abs[a - b]/20}], \ DisplayFunction \[Rule] Identity];\)\), "\[IndentingNewLine]", \(\(theFiber = \ fibra[theta[a], \ phi[a], \ {RGBColor[1, 0, 1], \ Thickness[0.01]}];\)\)}], "Input", CellLabel->"In[96]:="], Cell[BoxData[ \(\(Show[torus, \ theFiber, DisplayFunction \[Rule] $DisplayFunction];\)\)], "Input", CellLabel->"In[98]:="], Cell[BoxData[ \(We\ can\ agains\ see\ the\ animation\ of\ all\ the\ lifts . \ Note\ that\ the\ lift\ starts\ and\ finishes\ at\ the\ same\ fiber\ \ \((since\ the\ original\ curve\ is\ closed)\), \ but\ not\ in\ the\ same\ point . \ The\ difference\ beetwen\ the\ starting\ and\ ending\ point\ of\ the\ \ lift\ is\ the\ parallel\ displacement\ along\ the\ curve . Hence, \ we\ see\ that\ the\ holonomy\ and\ curvature\ of\ the\ connection\ are\ \ not\ trivial\ \(\((otherwise\ each\ lift\ would\ be\ closed)\)\(.\)\)\)], \ "Input", TextAlignment->Left, TextJustification->1, FontFamily->"Arial", FontSize->14, FontWeight->"Plain"], Cell[BoxData[ \(\(Table[ Show[torus, \ lift[m], theFiber, \ DisplayFunction \[Rule] $DisplayFunction, \ SphericalRegion \[Rule] True\ , \ PlotRange \[Rule] 1.8 {{\(-1\), 1}, \ {\(-1\), 1}, \ {\(-1.5\), 1.5}}], \ {m, \ 0, \ 2\ Pi, \ 2\ Pi/20}];\)\)], "Input", CellLabel->"In[99]:="] }, Open ]], Cell[CellGroupData[{ Cell["References", "Subtitle"], Cell[BoxData[ RowBox[{\(N . Blazic\), ",", \(S . Vukmirovic\), ",", StyleBox[\(Solutions\ of\ Yang - Mills\ equations\ on\ generalized\ Hopf\ bundles\), FontSlant->"Italic"], ",", \(J . Geom . Phys . 41\), ",", \(No .1 - 2\), ",", \(57 - 64\ \(\((2002)\)\(.\)\)\)}]], "Input", FontWeight->"Plain"], Cell[BoxData[ RowBox[{" ", RowBox[{\(T . G\[ODoubleDot]ckler\), ",", \(T . 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