arXiv subjects

David N. Pham

Publications and source records attributed to David N. Pham.

13 recordsLinked to original sources

arXiv2026

Twists, Codazzi Tensors, and the $6$-sphere

Let $(M,g,J,ω)$ be an almost Hermitian manifold. Given an automorphism $ψ\in \mathrm{Aut}(TM)$, the existing structure can be twisted to obtain a new almost Hermitian manifold $(M,g^ψ,J^ψ,ω^ψ)$. In the current paper, we study these $ψ$-twisted almost Hermitian structures with particular emphasis on questions regarding the integrability of $J^ψ$ and the Riemannian geometry of $g^ψ$. By studying the latter, we identity a certain class of $\mathrm{Aut}(TM)$ with nice transformation properties. We call these automorphisms $g$-\textit{Codazzi maps} because of their close relationship with Codazzi tensors. The aforementioned results are ultimately applied to the standard nearly Kähler structure on the $6$-sphere where we prove a nonintegrability result for the class of $g$-Codazzi maps.

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arXiv2026

On a Theorem of Wang for Complex Homogeneous Manifolds

In \cite{Wang1954}, Wang proved (among other things) a sufficiency result for a compact homogeneous manifold $G/H$ to admit a $G$-invariant complex structure. In this note, we give a new Lie theoretic proof of Wang's theorem which relies on nothing more than the familiar properties of the root space decomposition of a compact Lie group. It should be noted that the recent work of Ni and Wallach \cite{NiWallach2025} also revisits the aforementioned theorem of Wang (and others) and offers new Lie theoretic proofs as well. However, the approach of \cite{NiWallach2025} relies on such objects as Borel subalgebras, parabolic subalgebras, and Iwasawa decomposition which may be somewhat less familiar to the working differential geometer.

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arXiv2026

On the Chern-Ricci form of a twisted almost Kähler structure

Let $(M,g,J,ω)$ be an almost Kähler manifold. For any smooth function $f$ on $M$, one can associate an automorphism $ψ\in \mbox{Aut}(TM)$ for which the Kähler form is invariant. Using $ψ$, one can ``twist" the metric $g$ and almost complex structure $J$ to obtain a new almost Kähler structure $(g^ψ,J^ψ,ω)$ on $M$. Let $\widetilde{D}$ denote the Chern connection of $(g^ψ,J^ψ,ω)$ and let $K^{-1}$ denote the anti-canonical bundle of $(TM,J^ψ)$. In the current paper, we give an explicit formula for the local connection 1-form $α$ associated to the pair $(K^{-1},\widetilde{D})$. The Chern-Ricci form of $(g^ψ,J^ψ,ω)$ is then $ρ_{\widetilde{D}}=-dα$. We note that under certain conditions the aforementioned formula assumes a simpler form when applied to the calculation of $α$. We illustrate this with some examples.

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arXiv2025

On the non-existence of left-invariant hypercomplex structures on $SU(2)^{4n}$

Using elementary algebraic arguments, it is shown that $SU(2)^{m}:=SU(2)\times \cdots \times SU(2)$ ($m$ times) admits no left-invariant hypercomplex structures for all $m\ge 1$. This result answers (in a clear and easily accessible way) the question of whether every compact Lie group of dimension $4n$ admits a left-invariant hypercomplex structure. The aforementioned question has apparently been the source of some confusion in the recent literature.

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arXiv2024

Left invariant nearly pseudo-Kähler structures and the tangent lie group

Let $G$ be a Lie group, and let $(g,J)$ be a left invariant almost pseudo-Hermitian structure on $G$. It is shown that if $(g,J)$ is also nearly pseudo-Kähler, then the tangent bundle $TG$ (with its natural Lie group structure induced from $G$) admits a left-invariant nearly pseudo-Kähler structure.

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arXiv2024

A family of left-invariant SKT metrics on the exceptional Lie group $G_2$

For a complex manifold $(M,J)$, an SKT (or pluriclosed) metric is a $J$-Hermitian metric $g$ whose fundamental form $ω:=g(J\cdot,\cdot)$ satisfies the condition $\partial\overline{\partial}ω=0$. As such, an SKT metric can be regarded as a natural generalization of a Kähler metric. In this paper, the exceptional Lie group $G_2$ is equipped with a left-invariant integrable almost complex structure $\mathcal{J}$ via the Samelson construction and a 7-parameter family of $\mathcal{J}$-Hermitian metrics is constructed. From this 7-parameter family, the members which are SKT are calculated. The result is a 3-parameter family of left-invariant SKT metrics on $G_2$. As a special case, the aforementioned family of SKT metrics contains all bi-invariant metrics on $G_2$. In addition, this 3-parameter family of left-invariant SKT metrics are also invariant under the right action of a certain maximal torus $T$ of $G_2$. Conversely, it is shown that if $g$ is a left-invariant $\mathcal{J}$-Hermitian metric on $G_2$ such that $g$ is invariant under the right action of $T$ and for which $(g,\mathcal{J})$ is SKT, then $g$ must belong to this 3-parameter family of left-invariant SKT metrics.

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arXiv2024

Samelson complex structures for the tangent Lie group

It is shown that for any compact Lie group $G$ (odd or even dimensional), the tangent bundle $TG$ admits a left-invariant integrable almost complex structure, where the Lie group structure on $TG$ is the natural one induced from $G$. The aforementioned complex structure on $TG$ is inspired by Samelson's construction for even dimensional compact Lie groups.

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arXiv2023

Left-invariant Hermitian connections on Lie groups with almost Hermitian structures

Left-invariant Hermitian and Gauduchon connections are studied on an arbitrary Lie group $G$ equipped with an arbitrary left-invariant almost Hermitian structure $(\langle\cdot,\cdot\rangle,J)$. The space of left-invariant Hermitian connections is shown to be in one-to-one correspondence with the space $\wedge^{(1,1)}\mathfrak{g}^\ast\otimes \mathfrak{g}$ of left-invariant 2-forms of type (1,1) (with respect to $J$) with values in $\mathfrak{g}:=\mbox{Lie}(G)$. Explicit formulas are obtained for the torsion components of every Hermitian and Gauduchon connection with respect to a convenient choice of left-invariant frame on $G$. The curvature of Gauduchon connections is studied for the special case $G=H\times A$, where $H$ is an arbitrary $n$-dimensional Lie group, $A$ is an arbitrary $n$-dimensional abelian Lie group, and the almost complex structure is totally real with respect to $\mathfrak{h}:=\mbox{Lie}(H)$. When $H$ is compact, it is shown that $H\times A$ admits a left-invariant (strictly) almost Hermitian structure $(\langle\cdot,\cdot\rangle,J)$ such that the Gauduchon connection corresponding to the Strominger (or Bismut) connection in the integrable case is precisely the trivial left-invariant connection and, in addition, has totally skew-symmetric torsion. The almost Hermitian structure $(\langle\cdot,\cdot\rangle,J)$ on $H\times A$ is shown to satisfy the \textit{strong Kähler with torsion} condition. Furthermore, the affine line of Gauduchon connections on $H\times A$ with the aforementioned almost Hermitian structure is also shown to contain a (nontrivial) flat connection.

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arXiv2019

The Lie groupoid analogue of a symplectic Lie group

A symplectic Lie group is a Lie group with a left-invariant symplectic form. Its Lie algebra structure is that of a quasi-Frobenius Lie algebra. In this note, we identify the groupoid analogue of a symplectic Lie group. We call the aforementioned structure a \textit{$t$-symplectic Lie groupoid}; the "$t$" is motivated by the fact that each target fiber of a $t$-symplectic Lie groupoid is a symplectic manifold. For a Lie groupoid $\mathcal{G}\rightrightarrows M$, we show that there is a one-to-one correspondence between quasi-Frobenius Lie algebroid structures on $A\mathcal{G}$ (the associated Lie algebroid) and $t$-symplectic Lie groupoid structures on $\mathcal{G}\rightrightarrows M$. In addition, we also introduce the notion of a \textit{symplectic Lie group bundle} (SLGB) which is a special case of both a $t$-symplectic Lie groupoid and a Lie group bundle. The basic properties of SLGBs are explored.

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arXiv2017

$\frak{g}$-quasi-Frobenius Lie algebras

A Lie version of Turaev's $\overline{G}$-Frobenius algebras from 2-dimensional homotopy quantum field theory is proposed. The foundation for this Lie version is a structure we call a \textit{$\frak{g}$-quasi-Frobenius Lie algebra} for $\frak{g}$ a finite dimensional Lie algebra. The latter consists of a quasi-Frobenius Lie algebra $(\frak{q},β)$ together with a left $\frak{g}$-module structure which acts on $\frak{q}$ via derivations and for which $β$ is $\frak{g}$-invariant. Geometrically, $\frak{g}$-quasi-Frobenius Lie algebras are the Lie algebra structures associated to symplectic Lie groups with an action by a Lie group $G$ which acts via symplectic Lie group automorphisms. In addition to geometry, $\frak{g}$-quasi-Frobenius Lie algebras can also be motivated from the point of view of category theory. Specifically, $\frak{g}$-quasi Frobenius Lie algebras correspond to \textit{quasi Frobenius Lie objects} in $\mathbf{Rep}(\frak{g})$. If $\frak{g}$ is now equipped with a Lie bialgebra structure, then the categorical formulation of $\overline{G}$-Frobenius algebras given in \cite{KP} suggests that the Lie version of a $\overline{G}$-Frobenius algebra is a quasi-Frobenius Lie object in $\mathbf{Rep}(D(\frak{g}))$, where $D(\frak{g})$ is the associated (semiclassical) Drinfeld double. We show that if $\frak{g}$ is a quasitriangular Lie bialgebra, then every $\frak{g}$-quasi-Frobenius Lie algebra has an induced $D(\frak{g})$-action which gives it the structure of a $D(\frak{g})$-quasi-Frobenius Lie algebra.

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arXiv2014

Higher Affine Connections

For a smooth manifold $M$, it was shown in \cite{BPH} that every affine connection on the tangent bundle $TM$ naturally gives rise to covariant differentiation of multivector fields (MVFs) and differential forms along MVFs. In this paper, we generalize the covariant derivative of \cite{BPH} and construct covariant derivatives along MVFs which are not induced by affine connections on $TM$. We call this more general class of covariant derivatives \textit{higher affine connections}. In addition, we also propose a framework which gives rise to non-induced higher connections; this framework is obtained by equipping the full exterior bundle $\wedge^\bullet TM$ with an associative bilinear form $η$. Since the latter can be shown to be equivalent to a set of differential forms of various degrees, this framework also provides a link between higher connections and multisymplectic geometry.

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arXiv2013

Degenerate Monge-Type Hypersurfaces

In this note, we extend the notion of a Monge hypersurface from its roots in semi-Euclidean space to more general spaces. For the degenerate case, the geometry of these structures is studied using the Bejancu-Duggal method of screen distributions.

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arXiv2012

Generalized de Sitter Space in $n$-dimensional Minkowski Space

In this paper, we generalize the defining equation for de Sitter space by replacing the de Sitter radius with a function $f$ satisfying certain conditions; each resulting hypersurface is diffeomorphic to de Sitter space, and has a geometry (and causal character) which is controlled by the choice of $f$. Necessary and sufficient conditions are obtained for a hypersurface to be timelike, null, or spacelike in the generalized model; in the non-null case, the geometry is given by a warped product. Several examples of timelike, null, and spacelike hypersurfaces are presented. Lastly, we calculate the Ricci tensor and scalar curvature for a special family of 4-dimensional generalized de Sitter spaces.

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