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~$0$.
","abstract_html":"<p>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, ω-anti-self dual instantons; The third is pseudo-holomorphic curves.</p><p>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 CP3 and the twistor space of H4. </p><p>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, ω-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. </p><p>For the third perspective, I study pseudo-holomorphic curves in nearly K\\"ahler CP3. I construct a one-to-one correspondence between {\\it null torsion} curves in the nearly K\\"ahler CP3 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~$0$.</p>","abstract_has_math":true,"creators":["Xu, Feng"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Bryant, Robert"],"committee_chairs":[],"committee_members":[],"year":2008,"date_issued":"2008-05-14","date_published":"2008-05-14","updated_at":"2026-07-24T02:07:01Z","subjects":["Mathematics"],"languages":["en_US"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/10161/826","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Bryant, Robert"]},{"key":"dc:creator","label":"Author","values":["Xu, Feng"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2008-09-03T13:52:56Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2008-09-03T13:52:56Z"]},{"key":"dc:date.issued","label":"Date","values":["2008-05-14"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_US"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/10161/826"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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~$0$.
"]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Geometry of SU(3) Manifolds"]}]}],"canonical_facts":{"dc:contributor.advisor":["Bryant, Robert"],"dc:creator":["Xu, Feng"],"dc:date.accessioned":["2008-09-03T13:52:56Z"],"dc:date.available":["2008-09-03T13:52:56Z"],"dc:date.issued":["2008-05-14"],"dc:description.abstract":["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~$0$.
"],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["https://hdl.handle.net/10161/826"],"dc:language.iso":["en_US"],"dc:subject":["Mathematics"],"dc:title":["Geometry of SU(3) Manifolds"],"dc:type":["Dissertation"]},"updated_at":"2026-07-24T02:07:01Z"}