Isometric feature mapping (Isomap) is a promising manifold learning method. However, Isomap fails to work on data which distribute on clusters in a single manifold or manifolds. Many works have been done on extending Isomap to multi-manifolds learning. In this paper, we proposed a new multi-manifolds learning algorithm (M-Isomap) with the help of a general procedure. The new algorithm preserves...

In this paper,we study the Shafarevich conjecture for moduli space of polarized Calabi-Yau manifolds and obtain some results on the rigidity of families of Calabi-Yau manifolds. We use variation of Hodge structure and Higgs bundle to establish a criterion for rigidity and apply it to show some important families of Calabi-Yau manifolds are rigid,for examples: Lefschetz pencils of Calabi-Yau man...

Almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics are considered. A linear connection D is introduced such that the structure of these manifolds is parallel with respect to D. Of special interest is the class of the locally conformally equivalent manifolds to the hyper-Kähler manifolds of the investigated type and the case when the torsion of D is D-parallel. Curvature pro...

We show a number of applications to geometry of the study of cohomology algebras of various kinds of manifolds. The main tool is Hodge theory, and we use it to show that projective complex manifolds are more restricted topologically than compact Kähler manifolds. We also make explicit numerous constraints satisfied by cohomology algebras of compact Kähler manifolds, making them very non generic...

We show a rigidity theorem for the Seiberg–Witten invariants mod 2 families of spin 4-manifolds. A mechanism this also gives family version 10/8-type inequality. As an application, we prove existence non-smoothable topological 4-manifolds whose fiber, base space, and total space are smoothable as manifolds. These provide new examples $4$ -manifolds $M$ which inclusion maps $\operatorname {Diff}...

It is known that many Calabi-Yau manifolds form a connected web. The question of whether all Calabi–Yau manifolds form a single web depends on the degree of singularity that is permitted for the varieties that connect the distinct families of smooth manifolds. If only conifolds are allowed then, since shrinking two-spheres and three-spheres to points cannot affect the fundamental group, manifol...

We collect a list of known four-dimensional Fano manifolds and compute their quantum periods. This list includes all four-dimensional Fano manifolds of index greater than one, all fourdimensional toric Fano manifolds, all four-dimensional products of lower-dimensional Fano manifolds, and certain complete intersections in projective bundles.

The study of the local-global geometric topology of homology manifolds has a long history. Homology manifolds were introduced in the 1930s in attempts to identify local homological properties that implied the duality theorems satisfied by manifolds [23, 56]. Bing’s work on decomposition space theory opened new perspectives. He constructed important examples of 3-dimensional homology manifolds w...

In this paper we begin the study of polysymplectic manifolds, and of their relationship with PDE’s. This notion provides a generalization of symplectic manifolds which is very well suited for the geometric study of PDE’s with values in a smooth manifold. Some of the standard tools of analytical mechanics, such as the Legendre transformation and Hamilton’s equations, are shown to generalize to t...

In this paper we introduce harmonic Lagrangian submanifolds in general Kähler manifolds, which generalize special Lagrangian submanifolds in Calabi-Yau manifolds. We will use the deformation theory of harmonic Lagrangian submanifolds in Kähler manifolds to construct minimal Lagrangian torus in certain Kähler-Einstein manifolds with negative first Chern class.