For many years, I have worked on the The Goldberg Conjecture. In 1961, Prof. Samuel I. Goldberg published his prescient conjecture that

Every compact, symplectic, Einstein manifold is Kähler.

What do all these words mean ?

Kähler manifolds ( a class of complex manifolds which includes algebraic manifolds) possess a differential 2-form, $\omega$, which is non-degenerate ( i.e., $\omega\wedge\omega$ is never 0 ), and closed ( i.e., $\delta\omega=0$ ). Kähler manifolds are very special, even among complex manifolds. Y-T. Siu showed that a (real) 4 – dimensional, locally complex manifold is Kähler if and only if all of its odd-degree Betti numbers, $b_{2n+1}$, are even. Real manifolds which possess such a non-degenerate, closed differential 2-form, $\omega$, are called symplectic manifolds. They are necessarily even-dimensional. It used to be believed that closed, symplectic manifolds were always complex. However, in 1976, W. Thurston constructed a closed, symplectic 4-manifold which he showed could not be Kähler because its first Betti number, $b_{1}$, is 3. In 1983, I generalized Thurston’s example to an infinite family of closed, non-Kähler, symplectic manifolds. Unexpectedly, in 1995, R. Gompf, using a procedure called “logarithmic transformation,” constructed a plethora of closed, non-Kähler, symplectic 4-manifolds of almost any topology imaginable.

An n-dimensional Riemannnian manifold, M, is said to be Einstein if the its Ricci curvature 2-tensor, Ric(X,Y), is a constant multiple, sg, of its Riemannian metric tensor, g. That factor, s, is called the scalar curvature of M. If $n\geq3$, then the scalar curvature, s, is a constant on an Einstein manifold. Examples of Einstein manifolds include n-spheres and complex projective spaces. The Einstein condition is fundamental to general relativity.

The Seiberg-Witten equations are definable on any oriented, differentiable Riemannian 4-manifold, $(M,g)$. An extensive literature has deveoped and many stroing and surprsing results obtained.
The Goldberg Conjecture ( “Every compact, symplectic, Einstein manifold is Kähler”) is important in clarifying the Classification Conjecture for closed, differentiable 4-manifolds. This surprisingly difficult assertion has steadfastly resisted complete resolution. In 1987, K. Sekigawa used a very long, messy calculation to prove that Goldberg’s Conjecture is TRUE in all dimensions when the scalar curvature is non-negative ($s\geq0$). Independently, in 1984, I proved the same assertion in dimension four.

One may define the *-Ricci curvature tensor, $Ric^*$, incorporating the almost Hermitian structure, J , on an almost Hermitian manifold, in a manner analogous to that of the Ricci curvature tensor. An almost Hermitian manifold $(M^{2m},g,J)$ which satisfies $Ric^* = (\frac{s^*}{2m})g$ for the possibly varying *-scalar curvature functional, $s^*$, is called a (weakly) *-Einstein manifold. It is easy to verify that the norm squared, $\parallel\bigtriangledown J\parallel^2$ of the covariant differential of the almost complex structure, J, on an almost Kähler manifold is equal to $2(s^*-s)$. Moreover, the almost Kähler manifold $(M^{2m},g,J)$  is Kähler ifand only if $\parallel\bigtriangledown J\parallel^2=0$  We must pay particular attention to avoid assuming that $s^*$ is a global constant. A result of J. Armstrong implies that there is at least one point on a compact, four-dimensional Einstein almost Kähler manifold at which $s^*=s$. Therefore, if $s^*$ is assumed to be constant on the Einstein almost Kähler 4-manifold, then $s^*=s$ everywhere and the manifold must be Kähler. I tried to prove Goldberg’s Conjecture in the negative scalar curvature case ( $s^*<0$ ) using Seiberg-Witten invariants. The analysis is very delicate, and involves a topological invariant, $c_1(M)$, called the first Chern class of the manifold, M, which can be related to the total scalar curvature, i.e., the integral of $2(s^*+s)$ over the compact manifold. I use the easily established fact that a closed, Einstein, symplectic manifold is Kähler if and only if its constant scalar curvature, s, is equal to its possibly varying *-scalar curvature, $s^*$. (It is always true that $s\leq s^*$). In 1999, T. Oguro, Sekigawa and Yamada showed that a 4-dimensional compact almost Kähler manfold which is both Einstein and *-Einstein must be Kähler. However, it is very difficult to prove that a specific manifold is *-Einstein. Tedi Draghici and Vestislav Apostolov made great progress toward the Goldberg Conjecture and proved many related results. I feel that the Goldberg Conjecture is TRUE in dimension four and is FALSE in higher dimensions. The work of Catanese and LeBrun, who constructed an 8-dimensional manifold which has two Kähler-Einstein structures of opposite sign, gives hope to the prospect of constructing a higher-dimensional counterexample to the Goldberg Conjecture.

I am also presently trying to prove the Hard Lefschetz Property for compact, Einstein, symplectic manifolds:

CONJECTURE (Watson): The odd-degree Betti numbers of a compact, Einstein, symplectic 4-manifold, M, are even.

The verification of this conjecture would imply that the dimension, $b_{2}^{+}$, of the positive eigenspace of the intersection form, $Q$, is odd, making an analysis of the 4-dimensional Goldberg Conjecture using Seiberg-Witten invariants much easier.

In 1976, I defined almost Hermitian submersions to be Riemannian submersions between almost Hermitian manifolds which commute with the two involved almost Hermitian structures. This is a natural mathematical object to study if one is seeking Riemannian submersions with minimally embedded fibres as I was. In my doctoral thesis and its sequel, I showed that a surjective manifold map commutes with the Laplacian, $\Delta$, on 0-forms (resp. commutes with the codifferential, $\delta$, on 1-forms) if and only if the map is a Riemannian submersion with minimal fibres.

A natural question arises about almost Hermitian submersions. In 1980, Alfred Gray and Luis Hervella classified sixteen $U(n)$– invariant almost Hermitian structures on an almost Hermitian manifold, $(M^{2m},g,J)$. Most, but not all, of these sixteen classes are transferred from the total space to the base space of an almost Hermitian submersion. It is natural and interesting to ask if one assumes the relevant necessary $U(n)$– invariant structures on the base space and the fibres, what Gray-Hervella structure can be lifted up to the total space? In 1976, Lieven Vanhecke and I showed that the property of being almost semi-Kähler is lifted to the total space of an almost Hermitian submersion with almost semi-Kähler fibres and base space if and only if the fibres of the submersion are minimally embedded in the total space. Then, in the late nineties, I showed how the property of superminimality of the embedded fibre submanifolds, studied previously by R. Bryant and T. Friedrich, is related to the issue of lifting $U(n)$– invariant structures to the total space from the base space of an almost Hermitian submersion. There is a very weak $U(n)$– invariant structure called $G_1$. I showed that the property of being $G_1$ is lifted to the total space of an almost Hermitian submersion with $G_1$ fibres and $G_1$ base space if and only if the fibres of the submersion are superminimally embedded in the total space.

In 1998, I successfully proved that the four-dimensional total space of an almost Kähler submersion satisfies the Goldberg Conjecture without regard to the sign of the scalar curvature. This was accomplished by proving that the Einstein condition implies superminimal fibres which then implies that the almost complex structure on the total space is integrable (and thus the total space is Kähler), because the Gray-Hervella classes of almost Kähler and $G_1$ have only the Kähler manifolds in common. Compactness is not needed. This last remark is important, because John Armstrong has shown the possible existence of local counterexamples to the Goldberg Conjecture, while, in 1998, two Polish mathematicians produced a non-compact, Ricci-flat almost Kähler, non-Kähler manifold. Tedi Draghici, V. Apostolov and D. Kotschick proved that a compact, 4-dimensional almost Kähler manifold which satisfies the second curvature condition of Alfred Gray must be Kähler.

Later, I extended my work on Almost Kähler submersions to the case of a 6-dimensional total space. A few years ago, my colleague, Prof. T. Tshikuna-Matamba, in the Democratic Republic of the Congo, and I did some work (via e-mail) on placing the superminal fibre theorem into the context of almost contact mteric submersions. We prepared a manuscript, Superminimal Fibres in an Almost Contact Metric Submersion,” but have not yet had it published.

RECENT PUBLICATIONS

The Goldberg Conjecture is True for the Four-dimensional Total Space of an Almost Kähler Submersion. Journal of Geometry, 69 (2000). 215-226.

Superminimal Fibres in an Almost Hermitian Submersion, Boll. Unione Mat. Ital., 8 (2000), 159-172.

OTHER PUBLICATIONS

Almost Contact Metric 3 – Submersions, Intl. J. Math. Math. Sci., 7(1984), 667 – 688.

The Differential Geometry of Two Types of Almost Contact Metric Submersions, in The Mathematical Heritage of C.F. Gauss, ed. G. Rassias, World Scientific Publ. Co., Singapore, 1990, pp. 1 –  35.

New Examples of Strictly Almost Kähler Manifolds, Proc. Amer. Math. Soc.,  88(1983), 541 – 544.

Almost Hermitian Submersions, J. Differential Geometry, 11(1976), 147 – 165.

$\delta$ – Commuting Maps and Betti Numbers, Tôhoku Mathematics J., 27(1975), 137 – 152.

Manifold Maps Which Commute with the Laplacian, J. Differential Geometry, 8(1973), 89 -98.

A BOOK on some of my RESEARCHES

In Fall, 2004, the book “Riemannian Submersions and Related Topics,” by Maria Falcitelli, Stere Ianus and Anna Maria Pastore was published by World Scientific Publications. The book provides a research level exposition of this important field. Chapter 3, entitled “Almost Hermitian Submersions,” is devoted to the topic which I founded in 1976 (Almost Hermitian Submersions, Journal of Differential Geometry, vol. 11, pp. 147-165) and studied extensively in many of my subsequent publications. Chapter 4, entitled “Riemannian Submersions and Contact Metric Manifolds,” is devoted to Almost Contact Metric Submersions which I (and, independently, Prof. Dominic Chinea) created in 1984 ( The Differential Geometry of Two Types of Almost Contact Metric Submersions, in The Mathematical Heritage of C. F. Gauss (ed. G. Rassias) World Scientific Publ., pp. 827-861) and studied extensively in many of my subsequent publications. Chapter 5, entitled “Einstein Spaces and Riemannian Submersions,” addresses this important topic and elucidates the result I presented to the International Congress of Mathematicians in Berlin, Germany in August, 1998 entitled “The Goldberg Conjecture is True for the Four-dimensional Total Space of an Almost Hermitian Submersion.” Chapter 8 of the book, entitled “Applications of Riemannian Submersions in Physics,” devotes an entire section (8.1) to Gauge Fields, Instantons and Riemannian Submersions, developing the ansatz I reported in a paper entitled “Riemannian Submersions and Instantons,” delivered, by invitation, in 1980 to the Ninth International Conference on General Relativity and Gravitation(GR9), in Jena, German Democratic Republic, and published as “G, G’ – Riemannian Submersions and Nonlinear Gauge Field Equations in General Relativity,” in Global Analysis on Manifolds (G. Rassias, ed.), Teubner-Texte in Mathematics, vol. 57, pp. 324-349, 1983, Liebzig. The book, “Riemannian Submersions and Related Topics,” is available from amazon.com.