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Monday, 24 April 2017

A surface whose all points are hyperbolic

Consider the hyperbolic paraboloid, that is,  the surface S given by the graph of z=f(x,y)=x2y2. By the formula of the Gauss curvature for a surface which is a graph, we have K=fxxfyyf2xy(1+f2x+f2y)2=4(1+4x2+4y2)2<0. 


Then all points are negative. Consider the point p=(0,0,0)S, whose tangent plane is the plane z=0. Let c>0. Then the intersection of S with planes parallel to TpS is S{z=c}={(x,y,c):x2y2=c}={(x,y,c):(xc)2(yc)2=1},
that is, an ellipse, for each height c>0. Similarly, for c<0.

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