Topological restructuring of level surfaces of smooth functions on manifolds
No. 198, 1972
(Geometry)
India ink and color pencil on paper, 61x85 cm.
Windblown mountains and dunes fill this desert landscape, whose craggy peaks and valleys are highlighted by a distant source of illumination. Rough- edged rocks come together in places to create virtual cliffs that form borders to otherwise still and lifeless flats, while soft clouds float gently throughout. From a mathematical point of view, each smooth function on a smooth manifold fibers the latter into level hypersurfaces. As the value of the function increases, for instance, the corresponding level surface is deformed inside the manifold, changing its position. When the function's value passes through a critical point, the level surface changes its structure and topology. Thus neighboring level surfaces differ in that their critical points fall in the spaces between the hypersurfaces. In a sense, thematically, this picture reveals how level surfaces are restructured, especially where they are portrayed graphically as layers of clouds in the sky.
