  This paper is focused on geometric aspects of two particular types of
finite-variable reductions in the dispersionless Toda hierarchy. The reductions
are formulated in terms of "Landau-Ginzburg potentials" that play the role of
reduced Lax functions. One of them is a generalization of Dubrovin and Zhang's
trigonometric polynomial. The other is a transcendental function, the logarithm
of which resembles the waterbag models of the dispersionless KP hierarchy. They
both satisfy a radial version of the L\"owner equations. Consistency of these
L\"owner equations yields a radial version of the Gibbons-Tsarev equations.
These equations are used to formulate hodograph solutions of the reduced
hierarchy. Geometric aspects of the Gibbons-Tsarev equations are explained in
the language of classical differential geometry (Darboux equations, Egorov
metrics and Combescure transformations). Flat coordinates of the underlying
Egorov metrics are presented.
