Polar weighted homogeneous polynomials are special polynomials of real variables x_i, y_i, i = 1, ..., n with z_i = x_i + $\sqrt{-1}y_i$ which enjoy a "polar action". In many aspects, their behavior looks like that of complex weighted homogeneous polynomials. We study basic properties of hypersurfaces which are defined by polar weighted homogeneous polynomials.

For a two-dimensional surface M² in the four-dimensional Euclidean space E⁴ we introduce an invariant linear map of Weingarten type in the tangent space of the surface, which generates two invariants k and κ.

The condition k = κ = 0 characterizes the surfaces consisting of flat points. The minimal surfaces are characterized by the equality κ² - k = 0. The class of the surfaces with flat normal connection is characterized by the condition κ = 0. For the surfaces of general type we obtain a geometrically determined orthonormal frame field at each point and derive Frenet-type derivative formulas.

We apply our theory to the class of the rotational surfaces in E⁴, which prove to be surfaces with flat normal connection, and describe the rotational surfaces with constant invariants.

Let (X,0) be a germ of complex analytic normal variety, non-singular outside 0. An essential divisor over (X,0) is a divisorial valuation of the field of meromorphic functions on (X,0), whose center on any resolution of the germ is an irreducible component of the exceptional locus. The Nash map associates to each irreducible component of the space of arcs through 0 on X the unique essential divisor intersected by the strict transform of the generic arc in the component. Nash proved its injectivity and asked if it was bijective. We prove that this is the case if there exists a divisorial resolution π of (X,0) such that its reduced exceptional divisor carries sufficiently many π-ample divisors (in a sense we define). Then we apply this criterion to construct an infinite number of families of 3-dimensional examples, which are not analytically isomorphic to germs of toric 3-folds (the only class of normal 3-fold germs with bijective Nash map known before).

Let X be a projective variety of dimension n defined over the field of complex numbers and let L₁, ..., L_n-i be ample line bundles on X, where i is an integer with 0 ≤ i ≤ n. In this paper, first, we define some invariants called the ith sectional H-arithmetic genus, the ith sectional geometric genus and the ith sectional arithmetic genus of (X,L₁, ..., L_n-i). These are considered to be a generalization of invariants which have been defined in our previous papers. Moreover we investigate some basic properties of these, which are used in the second part and the third part of this work.

We completely classify the pairs of two smooth plane curves with double coverings between them. More precisely, we show that there exist no double coverings between two smooth plane curves except for several special cases.

Let F^m = (M, F) be a Finsler manifold and G be the Sasaki-Finsler metric on TM°. We show that the curvature tensor field of the Levi-Civita connection on (TM°, G) is completely determined by the curvature tensor field of Vrănceanu connection and some adapted tensor fields on TM°. Then we prove that (TM°, G) is locally symmetric if and only if F^m is locally Euclidean. Also, we show that the flag curvature of the Finsler manifold F^m is determined by some sectional curvatures of the Riemannian manifold (TM°, G). Finally, for any c ≠ 0 we introduce the c-indicatrix bundle IM (c) and obtain new and simple characterizations of F^m of constant flag curvature c by means of geometric objects on both IM (c) and (TM°, G).