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I was trying to replicate an image of a book, but I don't know how to increase the curvature on the curves. I've been done this so far:

documentclass[tikz, border=2pt]{standalone}

usepackage[utf8]{inputenc}

usepackage[brazilian]{babel}

usepackage{amssymb}

usepackage{tikz,tkz-euclide}

usetkzobj{all}

usepackage{xcolor}

usetikzlibrary{decorations.markings}



begin{document}

    begin{tikzpicture}[scale=3, mydot/.style={fill, circle, inner 

     sep=1.5pt}, decoration={markings, mark=at position 0.5 with 

     {arrow{latex}}}]



    draw[thick] (0,0) to[out=5,in=175, looseness=.8] (5,0);

    draw[thick] (0,0) to[out=-10,in=190, looseness=1.4] (5,0);

    draw[ultra thick] (0,0) to[out=-15,in=195, looseness=1.5] (5,0);

    draw[thick] (0,0) to[out=-25,in=205, looseness=1.6] (5,0);

    draw[thick] (0,0) to[out=-35,in=215, looseness=1.6] (5,0);

    draw[thick] (0,0) to[out=-45,in=225, looseness=1.7] (5,0);



    draw[thick] (1,-2) .. controls (2.3,-.567) and (2.5,.3) .. (2.3,1);



    draw[ultra thick,-latex,shorten >= 5pt] (2.3,-.567) to[out=45,in=45, 

    looseness=0] (2.8,.8);

    draw[ultra thick,-latex,shorten >= 5pt] (5.7,.7) to[out=190,in=80, 

    looseness=.8] (5,0);

    draw[ultra thick,-latex,shorten >= 5pt] (5,-1) to[out=120,in=1, 

    looseness=.7] (4,-.3);

    draw[ultra thick,-latex,shorten >= 5pt] (4,-2) to[out=120,in=1, 

    looseness=.7] (3.1,-1.6);



    node[mydot] at (0,0) {};

    node[mydot] at (5,0) {};



    node at (.9,-2.1) {{Large $X_p$}};

    node at (2.9,.85) {{Large $xi(p)$}};

    node at (5.85,.8) {{Large $partial M$}};

    node at (5.3,-1.1) {{Large $X_0=x$}};

    node at (4.1,-2.1) {{Large $X_t$}};

end{tikzpicture}

end{document}

The picture I want to replicate is that one bellow:

Using proper coordinates and plot command, smooth curves as shown in question can be reproduced. The format to use plot is:

draw[smooth] plot coordinates{<list of coordinates>};

A minimal working example:

documentclass[border=3mm]{standalone}

usepackage{tikz}



begin{document}

begin{tikzpicture}

fill (0,0) circle (2pt);

fill (5,0) circle (2pt);

draw[smooth,tension=0.7] plot coordinates{(0,0) (0.7,0.0) (1.8,0.2) (3.2,0.2) (4.3,0.0) (5,0)};

draw[thick,smooth,tension=0.7] plot coordinates{(0,0) (0.7,-0.1) (1.8,-0.4) (3.2,-0.4) (4.3,-0.1) (5,0)};

draw[smooth,tension=0.7] plot coordinates{(0,0) (0.7,-0.11) (1.8,-0.6) (3.2,-0.6) (4.3,-0.11) (5,0)};

draw[smooth,tension=0.7] plot coordinates{(0,0) (0.7,-0.12) (1.8,-0.7) (3.2,-0.7) (4.3,-0.12) (5,0)};

end{tikzpicture}

end{document}

Output:

Arguably, something like bend right might be better suited to produce some surface with constant mean curvature, but I am not claiming that this necessarily a closer reproduction of your screen shot. The main purpose of this answer is, however, to advertize relative positioning for the nodes and arrows.

documentclass[tikz, border=2pt]{standalone}

usepackage[utf8]{inputenc}

usepackage[brazilian]{babel}

usepackage{amssymb}

usepackage{tikz,tkz-euclide}

usetkzobj{all}

usepackage{xcolor}

usetikzlibrary{decorations.markings}



begin{document}

    begin{tikzpicture}[scale=3, mydot/.style={fill, circle, inner 

     sep=1.5pt}, decoration={markings, mark=at position 0.5 with 

     {arrow{latex}}},font=Large]



    foreach X in {-5,5,25,35,45}

    {draw[thick] (0,0) to[bend right=X] coordinate[pos=0.8] (auxX) (5,0);}

    draw[ultra thick] (0,0) to[bend right=15] coordinate[pos=0.4] (aux1) 

    coordinate[pos=0.7] (aux2) (5,0);



    draw[thick] (1,-2) node[below]{$X_p$} .. controls (2.3,-.567) and (2.5,.3) .. (2.3,1);





    draw[ultra thick,latex-] (aux1) -- ++(2,2) node[above]{

    $xi(p)$};

    draw[ultra thick,latex-] (aux2) to[bend right] ++ (1.2,-1.2) node[right]{$X_0=x$};

    draw[ultra thick,latex-] (aux35) to[bend right] ++ (1.2,-0.6)

    node[right]{$X_t$};



    node[mydot] (L) at (0,0) {};

    node[mydot] (R) at (5,0) {};

    draw[ultra thick,latex-] (R) to[bend left] ++ (1,0.5)

    node[right]{$partial M$};



end{tikzpicture}

end{document}

Just for the pleasure of using the Béziers curves.

I first printed the image of your book, having previously taken care to remove its greyish background.
Then, I measured some distances to position some points and some angles to place the tangents of the Béziers curves.

It is easier to place these tangents when using relative coordinates (see page 140-141 of manual 3.0.1a).

I composed these curves with an intermediate point placed in the middle by varying the ordinate in a foreach loop.
I placed an invisible node named (an) at each 0.2 of the second half of each path.

foreach y [count=n]in {.1,-.1,-.75,-.9,-1.14}{

draw [thin](0,0)

.. controls +(0:1) and +(180:1.5) .. (2.5,y) ..controls +(0:1.5) and +(180:1) .. (5,0)node[pos=.2](an){};

}

I drew Xo separately so I could thicken his line.

draw [ultra thick,name path=Xo](0,0)

.. controls +(0:1) and +(180:1.5) .. (2.5,-.5) ..controls +(0:1.5) and +(180:1) .. (5,0)node[pos=.4](a){};

To place the tangent, I calculated the intersection named ksi of the curve Xo and Xpand I still used the relative coordinates to draw this tangent.

% tangent

path[name intersections={of=Xp and Xo,by=ksi}];

draw[ultra thick,-Triangle,shorten >= 5pt] (ksi)--+(70:1) node[above ]{$xi(p)$};

The result and the complete code:

documentclass[tikz, border=5mm]{standalone}

usepackage[utf8]{inputenc}

usepackage[brazilian]{babel}

usepackage{amssymb}

usepackage{tikz,tkz-euclide}

usetkzobj{all}

usepackage{xcolor}

usetikzlibrary{shapes.geometric,intersections,arrows.meta}



begin{document}

    begin{tikzpicture}[scale=3, mydot/.style={fill, circle, inner 

     sep=1.5pt},

     every node/.style={font=Large},

     >={Latex[length=3mm]},

     ]

    node[mydot] at (0,0) {};

    node[mydot] at (5,0) (end){};



foreach y [count=n]in {.1,-.1,-.75,-.9,-1.14}{

draw [thin](0,0)

.. controls +(0:1) and +(180:1.5) .. (2.5,y) ..controls +(0:1.5) and +(180:1) .. (5,0)node[pos=.2](an){};

}



draw [ultra thick,name path=Xo](0,0) 

.. controls +(0:1) and +(180:1.5) .. (2.5,-.5) ..controls +(0:1.5) and +(180:1) .. (5,0)node[pos=.4](a){};



draw[<-,shorten <=5pt] (a)to[bend left]+(1,-.5)node[right]{ $X_0=x$};

draw[thick,name path=Xp] (.8,-1.6)node[below]{ $X_p$}

 .. controls +(50:1) and +(-110:.5) ..

 (2.1,-.5)

 ..controls +(70:.5) and +(-110:1.2)..(2.3,1);



% tangent

path[name intersections={of=Xp and Xo,by=ksi}];

draw[ultra thick,-Triangle,shorten >= 5pt] (ksi)--+(70:1) node[above ]{$xi(p)$};

% nodes

draw[thick,<-,shorten >= 5pt] (end) to[bend left] +(.5,.5)node[right]{$partial(M)$};

draw[<-] (a5)to[bend left]+(.5,-.5)node[right]{$X_t$};

end{tikzpicture}

end{document}

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