Browsing by Author "Bryant, Robert L"

In this dissertation, we study the geometry of Engel structures, which are 2-plane fields on 4-manifolds satisfying a generic condition, that are compatible with other geometric structures. A \emph{complex} Engel structure is an Engel 2-plane field on a complex surface for which the 2-planes are complex lines. A \emph{Lagrangian} Engel structure is an Engel 2-plane field on a symplectic 4-manifold for which the 2-planes are Lagrangian with respect to the symplectic structure. We solve the equivalence problems for complex Engel structures and Lagrangian Engel structures and use the resulting structure equations to classify homogeneous complex Engel structures and homogeneous Lagrangian Engel structures. This allows us to determine all compact, homogeneous examples.

For complex Engel structures, compact manifolds that support homogeneous complex Engel structures are diffeomorphic to $S^1\times SU(2)$ or quotients of $\mathbb{C}^2$, $S^1\times SU(2)$, $S^1\times G$ or $H$ by co-compact lattices, where $G$ is the connected and simply-connected Lie group with Lie algebra $\mathfrak{sl}_2(\mathbb{R})$ and $H$ is a solvable Lie group. For Lagrangian Engel structures, compact manifolds that support homogeneous Lagrangian Engel structures are diffeomorphic to quotients of one of a determined list of nilpotent or solvable 4-dimensional Lie groups by co-compact lattices.

This thesis is a study of B\"acklund transformations using geometric methods. A B\"acklund transformation is a way to relate solutions of two PDE systems. If such a relation exists for a pair of PDE systems, then, using a given solution of one system, one can generate solutions of the other system by solving only ODEs.

My contribution through this thesis is in three aspects.

First, using Cartan's Method of Equivalence, I prove the generality result: a generic rank-1 B\"acklund transformation relating a pair of hyperbolic Monge-Amp\`ere systems can be uniquely determined by specifying at most 6 functions of 3 variables. In my classification of a more restricted case, I obtain new examples of B\"acklund transformations, which satisfy various isotropy conditions.

Second, by formulating the existence problem of B\"acklund transformations as the integration problem of a Pfaffian system, I propose a method to study how a B\"acklund transformation relates the invariants of the underlying hyperbolic Monge-Amp\`ere systems. This leads to several general results.

Third, I apply the method of equivalence to study rank-$2$ B\"acklund transformations relating two hyperbolic Monge-Amp\`ere systems and partially classify those that are homogeneous. My classification so far suggests that those homogeneous B\"acklund transformations (relating two hyperbolic Monge-Amp\`ere systems) that are genuinely rank-2 are quite few.

I study differential geometry of 6-manifolds endowed with various $SU(3)$ structures from three perspectives. The first is special Lagrangian geometry; The second is pseudo-Hermitian-Yang-Mills connections or more generally, $\omega$-anti-self dual instantons; The third is pseudo-holomorphic curves.

For the first perspective, I am interested in the interplay between $SU(3)$ structures and their special Lagrangian submanifolds. More precisely, I study $SU(3)$-structures which locally support as `nice' special Lagrangian geometry as Calabi-Yau 3-folds do. Roughly speaking, this means that there should be a local special Lagrangian submanifold tangent to any special Lagrangian 3-plane. I call these $SU(3)$-structures {\it admissible}. By employing Cartan-K\"ahler machinery, I show that locally such admissible $SU(3)$-structures are abundant and much more general than local Calabi-Yau structures. However, the moduli space of the compact special Lagrangian submanifolds is not so well-behaved in an admissible $SU(3)$-manifold as in the Calabi-Yau case. For this reason, I narrow attention to {\it nearly Calabi-Yau} manifolds, for which the special Lagrangian moduli space is smooth. I compute the local generality of nearly Calabi-Yau structures and find that they are still much more general than Calabi-Yau structures. I also discuss the relationship between nearly Calabi-Yau and half flat $SU(3)$-structures. To construct complete or compact admissible examples, I study the twistor spaces of Riemannian 4-manifolds. It turns out that twistor spaces over self-dual Einstein 4-manifolds provide admissible and nearly Calabi-Yau manifolds. I also construct some explicit special Lagrangian examples in nearly K\"ahler $\mathbf{CP}^3$ and the twistor space of $H^4$.

For the second perspective, we are mainly interested in pseudo-Hermitian-Yang-Mills connections on nearly K\"ahler six manifolds. Pseudo-Hermitian-Yang-Mills connections were introduced by R. Bryant in \cite{BryantAlmCplx} to generalize Hermitian-Yang-Mills concept in K\"ahler geometry to almost complex geometry. If the $SU(3)$ structure is nearly K\"ahler, I show that pseudo-Hermitian-Yang-Mills connections (or, more generally, $\omega$-anti-self-dual instantons) enjoy many nice properties. For example, they satisfies the Yang-Mills equation and thus removable singularity results hold for such connections. Moreover, they are critical points of a Chern-Simons functional. I derive a Weitzenb\"ock formula for the deformation and discuss some of its application. I construct some explicit examples which display interesting singularities.

For the third perspective, I study pseudo-holomorphic curves in nearly K\"ahler $\mathbf{CP}^3$. I construct a one-to-one correspondence between {\it null torsion} curves in the nearly K\"ahler $\mathbf{CP}^3$ and contact curves in the K\"ahler $\mathbb{CP}^3$ (considered as a complex contact manifold). From this, I derive a Weierstrass formula for all {\it null torsion} curves by employing a result of R. Bryant in \cite{BryantS^4}. In this way, I classify all pseudo-holomorphic curves of genus~$0$.

A GL(2,R)-structure on a smooth manifold of dimension n+1 corresponds to a distribution of non-degenerate rational normal cones over the manifold. Such a structure is called k-integrable if there exist many foliations by submanifolds of dimension k whose tangent spaces are spanned by vectors in the cones.

This structure was first studied by Bryant for n=3 and k=2. The work included here (n=4 and k=2,3) was suggested by Ferapontov, et al., who showed that the cases (n=4,k=2) and (n=4, k=3) can arise from integrability of second-order PDEs via hydrodynamic reductions.

Cartan--Kahler analysis for n=4 and k=3 leads to a complete classification of local structures into 54 equivalence classes determined by the value of an essential 9-dimensional representation of torsion for the GL(2,R)-structure. These classes are described by the factorization root-types of real binary octic polynomials. Each of these classes must arise from a PDE, but the PDEs remain to be identified.

Also, we study the local problem for n >= 5 and k=2,3 and conjecture that similar classifications exist for these cases; however, the interesting integrability results are essentially unique to degree 4. The approach is that of moving frames, using Cartan's method of equivalence, the Cartan--Kahler theorem, and Cartan's structure theorem.

The fundamental point-wise invariant of a Riemannian manifold $(M, g)$ is the Riemann curvature tensor. Many special types of Riemannian manifolds can be characterized by conditions on the Riemann curvature tensor and tensor fields derived from it. Examples include Einstein manifolds and conformally flat manifolds.

Here we restrict ourselves to three dimensions and explore the Remannian manifolds that arise when imposing conditions on the irreducible components of the first covariant derivative of the Riemann curvature tensor. Specifically, we look at an irreducible component of the covariant derivative which takes the form of a traceless symmetric $(0,3)$-tensor field. We classify the local and global structure of manifolds where this tensor field vanishes.

In this thesis we analyze families of minimal Lagrangian submanifolds of complex projective 3-space CP3 whose fundamental cubic forms satisfy geometrically natural conditions at every point, namely that their fundamental cubic form be preserved by a proper subgroup of SO(3). There is a classification of SO(3)-stabilizer types of such fundamental cubics, which shows there are precisely five families of such cubic forms: Those with stabilizers contained in SO(2), A4, S3, Z2, and Z3. We use the method of moving frames, along with exterior differential systems techniques to prove existence of minimal Lagrangian submanifolds with each stabilizer type. We also attempt to integrate the resulting structure equations to give explicit examples of each.

I use the methods of exterior differential systems and the moving frame to study two geometric structures in seven dimensions related to $G_2$-geometry, and linked by the idea of special torsion. The torsion tensor of a geometric structure is a basic first-order invariant of the structure, and both of the geometries I study have special torsion, meaning that the image of their torsion tensor is constrained to lie in a smaller than usual subset.

In part 1, I study quadratic closed $G_2$-structures. A closed $G_2$-structure on a 7-manifold $M$ is given by a closed nondegenerate 3-form $\varphi$, and the quadratic condition, introduced by Bryant, says that $\varphi$ satisfies one of a particular natural one-parameter family of second order equations. The torsion tensor associated to a closed $G_2$-structure $\varphi$ takes values in $\mathfrak{g}_2$, and I study the cases where the image of this map lies in an exceptional $G_2$-orbit. A closed $G_2$-structure $\varphi$ induces a metric, and I give a classification of closed $G_2$-structures with conformally flat induced metric.

In part 2, I study $G_2$-structures endowed with a distribution of calibrated planes. In this situation there is an induced $SO(4)$-structure, and I invesitigate the cases where the $G_2$-structure is torsion-free and the induced $SO(4)$-structure has torsion tensor taking values in an irreducible $SO(4)$-module. Additionally, I give a classification of $SO(4)$-structures with invariant torsion, meaning that their torsion tensor takes values in a direct sum of trivial $SO(4)$-modules.