Associative submanifolds of a G2 manifold

Calibrated Manifolds and Gauge Theory

By a theorem of Mclean, the deformation space of an associative submanifold Y of an integrable G_2 manifold (M,ϕ) can be identified with the kernel of a Dirac operator D:Ω^{0}(ν) -->Ω^{0}(ν) on the normal bundle νof Y. Here, we generalize this to the non-integrable case, and also show that the deformation space becomes smooth after perturbing it by natural parameters, which corresponds to moving Y through `pseudo-associative' submanifolds. Infinitesimally, this corresponds to twisting the Dirac operator D --> D_{A} with connections A of ν. Furthermore, the normal bundles of the associative sub…

Lagrangian-type submanifolds of Spin(7) manifolds and their deformations

In an earlier paper we showed that the space of deformations of a smooth, compact, orientable Harvey-Lawson submanifold HL in a G2 manifold M can be identified with the direct sum of the space of smooth functions and closed 2-forms on HL. In that paper, we also introduced a new class of Lagrangian-type 4-dimensional submanifolds inside G2 manifolds, called them RS submanifolds, and proved that the space of deformations of a smooth, compact, orientable RS submanifold in a G2 manifold M can be identified with closed 3-forms on RS. In this short note, we define a new class of Lagrangian-type 4-di…

Deformations of Lagrangian Type Submanifolds inside G2 manifolds

3-dimensional Harvey Lawson submanifolds were introduced in an earlier paper by Akbulut-Salur, as examples of Lagrangian-type manifolds inside G2 manifold. In this paper, we first show that the space of deformations of a smooth, compact, orientable Harvey-Lawson submanifold HL in a G2 manifold M can be identified with the direct sum of the space of smooth functions and closed 2-forms on HL. We then introduce a new class of Lagrangian-type 4-dimensional submanifolds inside G2, call them RS submanifolds and prove that the space of deformations of a smooth, compact, orientable RS submanifold in a…

Surfaces, submanifolds, and aligned Fox reimbedding in non-Haken 3-manifolds

Understanding non-Haken 3-manifolds is central to many current endeavors in 3-manifold topology. We describe some results for closed orientable surfaces in non-Haken manifolds, and extend Fox's theorem for submanifolds of the 3-sphere to submanifolds of general non-Haken manifolds. In the case where the submanifold has connected boundary, we show also that the boundary-connected sum decomposition of the submanifold can be aligned with such a structure on the submanifold's complement.

Einstein流形中的极小子流形与稳定性

Minimal submanifolds and stability in Einstein manifolds

本文计算了复Einstein空间极小子流形的指标与零度，研究其稳定性并提出不稳定性准则，同时计算了Berger球面拉普拉斯算子的高阶特征值并给出算法。

The relations among two transversal submanifolds and global manifold

In Riemann geometry, the relations among two transversal submanifolds and global manifold are discussed. By replacing the normal vector of a submanifold with the tangent vector of another submanifold, the metric tensors, Christoffel symbols and curvature tensors of the three manifolds are linked together. When the inner product of the two tangent vectors vanishes, some corollaries of these relations give the most important second fundamental form and Gauss-Codazzi equation in the conventional submanifold theory. As a special case, the global manifold is Euclidean is considered. It is pointed o…