Wen-ling Huang

Adjacency Preserving Maps on Hermitian Matrices

Huas fundamental theorem of the geometry of hermitian matrices characterizes bijective mapsonthespaceofall n×nhermitianmatricespreservingadjacencyinbothdirections. Theproblem of possible improvements has been open for a while. There are three natural problems here. Do we need the bijectivity assumption? Can we replace the assumption of preserving adjacency in both directions by the weaker assumption of preserving adjacency in one direction only? Can we obtain such a characterization for maps acting between the spaces of hermitian matrices of different sizes? We answer all three questions for the complex hermitian matrices, thus obtaining the optimal structural result for adjacency preserving maps on hermitian matrices over the complex field.

Adjacency preserving mappings of invariant subspaces of a null system

In the space Ir of invariant r-dimensional subspaces of a null system in (2r + l)-dimensional projective space, W.L. Chow characterized the basic group of transformations as all the bijections : I, -+ Ir, for which both W and -1 preserve adjacency. In the present paper we show that the two conditions p: I, -+ I, is a surjection and W preserves adjacency are sufficient to characterize the basic group. At the end of this paper we give an application to Lie geometry.

Characterization of the Transformation Group of the Space of a Null System

In the space Ir of the invariant r-dimensional subspaces of a null system in (2r +1)-dimensional projective space, W.L. Chow characterized the basic group of transformations of Ir as all the transformations φ: Ir → Ir which are bijective and such that φ and φ−1 preserve adjacency. In the present paper we examine arbitrary mappings φ of Ir which satisfy the two conditions: 1. φ preserves adjacency. 2. For any a ∈ Ir there exists b ∈ Ir such that aφ ∩ bφ = ø.

On the Fundamental Theorems of the Geometries of Symmetric Matrices

In the present paper the relation between the fundamental theorem of Sn(F) and the fundamental theorem ofP Sn(F)is studied geometrically.

Positivity of the Bondi mass in Bondi's radiating spacetimes

We find two conditions related to the {\it news functions} of the Bondis radiating vacuum spacetimes. We provide a complete proof of the positivity of the Bondi mass by using Schoen-Yaus method under one condition and by using Wittens method under another condition.

Geraden-Abbildungen, die den Flächeninhalt 1 von Dreiecken erhalten

Zusammenfassung und VorwortZu jeder Geometrie, in der Geraden definiert sind, gehört eine Geradengeometrie, in der interessante Fragestellungen auftauchen. Die klassische Frage in der Liniengeometrie Plückers lautet: “Gegeben sei ein dreidimensionaler projektiver Raum. Wie lauten diejenigen bijektiven Geradenabbildungen, für die sich zwei Geraden genau dann schneiden, wenn sich deren Bilder schneiden?” Eine solche Abbildung ist entweder induziert durch eine Kollineation des projektiven Raumes oder eine Dualität. W. L. Chow und H. Brauner haben in [5] und [4] diese Fragestellung verallgemeinert. W. Benz hat in [1] die klassische Frage Plückers im ℝn fürn ≥ 3 beantwortet. J. Lester sucht in [6] diejenigen Geradenabbildungen des dreidimensionalen euklidischen Raumes, die den Abstand 1 von Geraden in beiden Richtungen erhalten.In meiner Arbeit bestimme ich diejenigen Geradenabbildungenπ desn-dimensionalen reellen euklidischen Raumes, die den Inhalt 1 von Dreiecken unverändert lassen (n ≥ 2). Dieser Frage hat sich auch schon Frau Lester gewidmet ([8]), aber unter der zusätzlichen, starken Voraussetzung, daßπ bijektiv sein soll. Ich werde zeigen, daß diese Forderung unnötig ist; der Nachweis der Surjektivität beinhaltet die eigentliche Schwierigkeit.