The $K$-rings of non-singular complex projective varieties as well as quasi-toric manifolds were described in terms of generators and relations in earlier work of the author with V. Uma. In this paper we obtain a similar description for the more general class of torus manifolds with locally standard torus action and orbit space a homology polytope, which includes the class of all smooth complete complex toric varieties.

We classify Fano fivefolds of index two which are blow-ups of smooth manifolds along a smooth center.

An immersed umbilic-free hypersurface in the unit sphere is equipped with three Möbius invariants, namely, the Möbius metric, the Möbius second fundamental form and the Möbius form. The fundamental theorem of Möbius submanifolds geometry states that a hypersurface of dimension not less than three is uniquely determined by the Möbius metric and the Möbius second fundamental form. A Möbius isoparametric hypersurface is defined by two conditions that it has vanishing Möbius form and has constant Möbius principal curvatures. It is well-known that all Euclidean isoparametric hypersurfaces are Möbius isoparametrics, whereas the latter are Dupin hypersurfaces. In this paper, combining with previous results, a complete classification for all Möbius isoparametric hypersurfaces in the unit six-sphere is established.

We extend the concept of intrinsic ultracontractivity to non-symmetric semigroups and prove the intrinsic ultracontractivity of the Dirichlet semigroups of non-symmetric second order elliptic operators in bounded Lipschitz domains.

We construct invariant complex structures of a compact 3-symmetric space by means of the canonical almost complex structure of the underlying manifold and some involutions of a Lie group. Moreover, by making use of graded Lie algebras and some invariant structures of affine symmetric spaces, we classify half dimensional, totally real and totally geodesic submanifolds of a compact 3-symmetric space with respect to each invariant complex structure.

We present a class of nonlinear differential equations of second Painlevé type. These equations, with a single exception, admit the quasi-Painlevé property along a rectifiable curve, that is, for general solutions, every movable singularity defined by a rectifiable curve is at most an algebraic branch point. Moreover we discuss the global many-valuedness of their solutions. For the exceptional equation, by the method of successive approximation, we construct a general solution having a movable logarithmic branch point.

In a complex projective space, we distinguish hypersurfaces of type $({\rm A}_1)$ from hypersurfaces of type $({\rm A}_2)$ in terms of the cardinality of congruence classes of their extrinsic geodesics.