Gdtyjr

I'm quite new to TikZ, so I would love to get some help with the drawing! What I intended to draw was not exactly what is shown in the picture, as that is probably not possible. I only want to draw a similar photo within the boundaries of TikZ. That is, the surfaces themselves don't have to curve exactly like the ones in the photo.

Here is the code

documentclass[border=5mm,tikz]standalone
usepackageamssymb
usepackagepgfplots
usepgfplotslibrarypatchplots
usetikzlibrarypatterns, positioning, arrows
pgfplotssetcompat=1.8
usepackagetikz


begindocument

begintikzpicture

 % Functions i
 path[->] (0.8, 0) edge [bend right] node[left, xshift=-2mm] $phi_i$ (-1, -2.9);
 draw[white,fill=white] (0.06,-0.57) circle (.15cm);
 path[->] (-0.7, -3.05) edge [bend right] node [right, yshift=-3mm] $phi^-1_i$ (1.093, -0.11);
 draw[white, fill=white] (0.95,-1.2) circle (.15cm);

 % Functions j
 path[->] (5.8, -2.8) edge [bend left] node[midway, xshift=-5mm, yshift=-3mm] $phi^-1_j$ (3.8, -0.35);
 draw[white, fill=white] (4,-1.1) circle (.15cm);
 path[->] (4.2, 0) edge [bend left] node[right, xshift=2mm] $phi_j$ (6.2, -2.8);
 draw[white, fill=white] (4.54,-0.12) circle (.15cm);

 % Manifold
 draw[smooth cycle, tension=0.4, fill=white, pattern color=brown, pattern=north west lines, opacity=0.7] plot coordinates(2,2) (-0.5,0) (3,-2) (5,1) node at (3,2.3) $M$;

 % Help lines
 %draw[help lines] (-3,-6) grid (8,6);

 % Subsets
 draw[smooth cycle, pattern color=orange, pattern=crosshatch dots] plot coordinates (1,0) (1.5, 1.2) (2.5,1.3) (2.6, 0.4); 
 node [label=[label distance=-0.3cm, xshift=-2cm, fill=white]:$U_i$] ;
 draw[smooth cycle, pattern color=blue, pattern=crosshatch dots] 
 plot coordinates (4, 0) (3.7, 0.8) (3.0, 1.2) (2.5, 1.2) (2.2, 0.8) (2.3, 0.5) (2.6, 0.3) (3.5, 0.0); node [label=[label distance=-0.8cm, xshift=.75cm, yshift=1cm, fill=white]:$U_j$] ;

 % First Axis
 draw[thick, ->] (-3,-5) -- (0, -5) node [label=above:$phi_i(U_i)$] ;
 draw[thick, ->] (-3,-5) -- (-3, -2) node [label=right:$mathbbR^m$] ;

 % Arrow from i to j
 draw[->] (0, -3.85) -- node[midway, above]$psi_ij$ (4.5, -3.85);

 % Second Axis
 draw[thick, ->] (5, -5) -- (8, -5) node [label=above:$phi_j(U_j)$] ;
 draw[thick, ->] (5, -5) -- (5, -2) node [label=right:$mathbbR^m$] ;

 % Sets in R^m
 draw[white, pattern color=orange, pattern=crosshatch dots] (-0.67, -3.06) -- +(180:0.8) arc (180:270:0.8);
 fill[even odd rule, white] [smooth cycle] plot coordinates(-2, -4.5) (-2, -3.2) (-0.8, -3.2) (-0.8, -4.5) (-0.67, -3.06) -- +(180:0.8) arc (180:270:0.8);
 draw[smooth cycle] plot coordinates(-2, -4.5) (-2, -3.2) (-0.8, -3.2) (-0.8, -4.5);
 draw (-1.45, -3.06) arc (180:270:0.8);

 draw[white, pattern color=blue, pattern=crosshatch dots] (5.7, -3.06) -- +(-90:0.8) arc (-90:0:0.8);
 fill[even odd rule, white] [smooth cycle] plot coordinates(7, -4.5) (7, -3.2) (5.8, -3.2) (5.8, -4.5) (5.7, -3.06) -- +(-90:0.8) arc (-90:0:0.8);
 draw[smooth cycle] plot coordinates(7, -4.5) (7, -3.2) (5.8, -3.2) (5.8, -4.5);
 draw (5.69, -3.85) arc (-90:0:0.8);

endtikzpicture

enddocument

I don't get anywhere near the figure I need to make.