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I would like to draw the following figure:

To do this I have used the following codes :

documentclass[10pt]article
usepackagepgf,tikz
usetikzlibraryarrows
pagestyleempty
begindocument

begintikzpicture[line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm]
draw [rotate around=15.05:(6.07,0.75)] (6.07,0.75) ellipse (1.11cm and 0.56cm);
draw [rotate around=-13.74:(8.9,0.78)] (8.9,0.78) ellipse (1.07cm and 0.53cm);
draw (4.58,2.02)-- (11,2);
draw (11,2)-- (10.48,-1);
draw (10.48,-1)-- (4,-1);
draw (4,-1)-- (4.58,2.02);
draw [shift=(6.76,0.91)] plot[domain=1.71:4.85,variable=t](1*0.43*cos(t r)+0*0.43*sin(t r),0*0.43*cos(t r)+1*0.43*sin(t r));
draw [shift=(8.25,0.9)] plot[domain=-1.6:1.54,variable=t](1*0.42*cos(t r)+0*0.42*sin(t r),0*0.42*cos(t r)+1*0.42*sin(t r));
draw [rotate around=-3.74:(8.09,7.47)] (8.09,7.47) ellipse (3.48cm and 1.89cm);
draw [rotate around=13.37:(7.03,7.4)] (7.03,7.4) ellipse (1.2cm and 0.59cm);
draw [rotate around=-18.43:(8.48,7.38)] (8.48,7.38) ellipse (1.23cm and 0.59cm);
draw [->] (8.96,6.46) -- (8.8,1.62);
draw [->] (6.7,6.38) -- (6.46,1.62);
draw [->] (7.3,0.98) -- (7.76,0.98);


draw (9.54,9.82) node $M$;
draw (6.8,7.3) node $U_i$;
draw (8.8,7.3) node $U_j$;
draw (6.24,4.34) node $psi_i$;
draw (9.52,4.2) node $psi_j$;
draw (8.94,-1.3) node $mathbfR^n$;
draw (7.6,0) node $psi_jcirc psi_i^-1$;

endtikzpicture
enddocument

It produces:

How can I shade this figure?

There are two basic tricks that allow you to fill the area bounded by two different curves/contours:

1. clip against one curve and fill the other;


2. use even odd rule.

And there are combinations of the two and other possibilities. This answer focuses on possibility 1. Then there is the question how on could recycle curves for the fill. Out of several possibilities, this answer will utilize the use path trick in the first part and insert path in the second path.

The first path modifies your code such as to shade the correct (?) areas.

documentclass[10pt]article
usepackagetikz
usetikzlibraryarrows
makeatletter % https://tex.stackexchange.com/a/38995/121799
tikzset
 use path/.code=pgfsyssoftpath@setcurrentpath#1

makeatother

pagestyleempty
begindocument

begintikzpicture[line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm]
draw [rotate around=15.05:(6.07,0.75),save path=pathA] (6.07,0.75) ellipse (1.11cm and 0.56cm);
draw [rotate around=-13.74:(8.9,0.78),save path=pathB] (8.9,0.78) ellipse (1.07cm and 0.53cm);
draw (4.58,2.02)-- (11,2);
draw (11,2)-- (10.48,-1);
draw (10.48,-1)-- (4,-1);
draw (4,-1)-- (4.58,2.02);
draw [shift=(6.76,0.91)] plot[domain=1.71:4.85,variable=t](1*0.43*cos(t r)+0*0.43*sin(t r),0*0.43*cos(t r)+1*0.43*sin(t r));
draw [shift=(8.25,0.9)] plot[domain=-1.6:1.54,variable=t](1*0.42*cos(t r)+0*0.42*sin(t r),0*0.42*cos(t r)+1*0.42*sin(t r));
draw [rotate around=-3.74:(8.09,7.47)] (8.09,7.47) ellipse (3.48cm and 1.89cm);
draw [save path=pathC,rotate around=13.37:(7.03,7.4)] (7.03,7.4) ellipse (1.2cm and 0.59cm);
draw [save path=pathD,rotate around=-18.43:(8.48,7.38)] (8.48,7.38) ellipse (1.23cm and 0.59cm);
draw [->] (8.96,6.46) -- (8.8,1.62);
draw [->] (6.7,6.38) -- (6.46,1.62);
draw [->] (7.3,0.98) -- (7.76,0.98);


draw (9.54,9.82) node $M$;
draw (6.8,7.3) node $U_i$;
draw (8.8,7.3) node $U_j$;
draw (6.24,4.34) node $psi_i$;
draw (9.52,4.2) node $psi_j$;
draw (8.94,-1.3) node $mathbfR^n$;
draw (7.6,0) node $psi_jcirc psi_i^-1$;
beginscope
clip[use path=pathA];
path[fill=blue,shift=(6.76,0.91)] plot[domain=1.71:4.85,variable=t](1*0.43*cos(t r)+0*0.43*sin(t r),0*0.43*cos(t r)+1*0.43*sin(t r))
-- ++ (1,0) |- cycle;
endscope
beginscope
clip[use path=pathB];
path[fill=blue,shift=(8.25,0.9)] plot[domain=-1.6:1.54,variable=t](1*0.42*cos(t r)+0*0.42*sin(t r),0*0.42*cos(t r)+1*0.42*sin(t r))
-- ++ (-1,0) |- cycle;
endscope
clip[use path=pathC];
fill[blue,use path=pathD];
endtikzpicture
enddocument

However, I am wondering if you are willing to consider an arguably simpler code yielding a similar picture. Advantages include more relative positioning such that you can move complete parts around without having to redo all coordinates.

documentclass[10pt]article
usepackagetikz
usetikzlibraryarrows,patterns
pagestyleempty
begindocument

begintikzpicture[line cap=round,line join=round,>=triangle
45,x=1.0cm,y=1.0cm,standard ellipse around/.style args=#1 rotated by #2%
insert path=[rotate around=#2:#1] #1 circle[x radius=1.2cm,y radius=0.6cm]]
 beginscope[yshift=6.5cm]
 draw (0,0) circle[x radius=3.5cm,y radius=1.9cm];
 node at (2,2.3) $M$;
 draw (-0.7,-0.2) node[left] (Ui) $U_i$
 [standard ellipse around=(-0.7,-0.2) rotated by 15];
 draw (0.7,-0.2) node[right] (Uj) $U_i$
 [standard ellipse around=(0.7,-0.2) rotated by -15];
 clip[standard ellipse around=(0.7,-0.2) rotated by -15];
 path[pattern=north east lines, 
 standard ellipse around=(-0.7,-0.2) rotated by 15];
 endscope
 beginscope[local bounding box=b]
 beginscope[xshift=-4mm,local bounding box=bl]
 draw[clip,standard ellipse around=(-1.2,0) rotated by 15];
 draw[pattern=north east lines,standard ellipse around=(0,0) rotated by -15];
 endscope
 beginscope[xshift=4mm,local bounding box=br]
 draw[clip,standard ellipse around=(1.2,0) rotated by -15];
 draw[pattern=north east lines,standard ellipse around=(0,0) rotated by 15];
 endscope
 draw [->] (bl) -- (br) node[midway,below=8mm]$psi_jcirc psi_i^-1$;
 endscope
 draw[->] ([yshift=-0.5cm]Ui.south) -- ([yshift=2mm]bl.north-|Ui.south)
 node[midway,left]$psi_i$;
 draw[->] ([yshift=-0.5cm]Uj.south) -- ([yshift=2mm]br.north-|Uj.south)
 node[midway,right]$psi_j$;
 draw ([xshift=-1.5cm,yshift=-1cm]b.south west)
 -- ([xshift=-1cm,yshift=1cm]b.north west)
 -- ([xshift=1.5cm,yshift=1cm]b.north east)
 -- ([xshift=1cm,yshift=-1cm]b.south east) -- cycle;
endtikzpicture
enddocument