scholarly journals Projective Connections and Projective Transformations

1957 ◽  
Vol 12 ◽  
pp. 1-24 ◽  
Author(s):  
Noboru Tanaka
Keyword(s):  
Affine Transformations ◽  
Connected Manifold ◽  
Projective Transformations

The main purpose of the present paper is to establish a theorem concerning the relation between the group of all projective transformations on an affinely connected manifold and the group of all affine transformations.

Projective and affine transformations of a complex symmetric connection

1994 ◽  
Vol 50 (2) ◽  
pp. 337-347 ◽  
Author(s):  
Novica Blažić ◽  
Neda Bokan
Keyword(s):  
Complex Manifold ◽  
Projective Transformation ◽  
Compact Complex Manifold ◽  
Affine Transformations ◽  
Projective Transformations ◽  
Chern Numbers ◽  
Arbitrary Complex ◽  
Riemannian Connection ◽  
Symmetric Connection ◽  
Complex Symmetric

Let M be a compact complex manifold and ∇ an arbitrary complex (not necessarily Riemannian) connection. In this paper we study the relation between the geometry of (M, ∇) and the topology of M, that is, we are interested in the following problem: To what extent does the topology of M determine the relations between the group of holomorphically projective transformations, the group of projective transformations and the group of affine transformations on M? Under assumptions on the Ricci-type tensors of ∇ and Chern numbers of M we show that a holomorphically projective transformation and a projective transformation are in fact affine transformations on M. A family of interesting examples of connections of this kind are constructed. Also, the case when M is a Kähler manifold is studied.

scholarly journals Note on the Group of Affine Transformations of an Affinely Connected Manifold

1955 ◽  
Vol 8 ◽  
pp. 71-81 ◽  
Author(s):  
Jun-Ichi Hano ◽  
Akihiko Morimoto
Keyword(s):  
Lie Group ◽  
Present Note ◽  
Affine Transformations ◽  
Connected Manifold

The purpose of the present note is to reform Mr. K. Nomizu’s result on the group of all affine transformations of an affinely connected manifold. We shall prove the following. THEOREM. The group of all affine transformations of an affinely connected manifold is a Lie group.

The Standard Metric

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Affine Transformation ◽  
Convex Sets ◽  
Finite Dimension ◽  
Affine Subspace ◽  
Affine Transformations ◽  
Real Vector ◽  
Euclidean Metric ◽  
Affine Hyperplane ◽  
Tits Building ◽  
Convex Closure

This chapter presents some results about groups generated by reflections and the standard metric on a Bruhat-Tits building. It begins with definitions relating to an affine subspace, an affine hyperplane, an affine span, an affine map, and an affine transformation. It then considers a notation stating that the convex closure of a subset a of X is the intersection of all convex sets containing a and another notation that denotes by AGL(X) the group of all affine transformations of X and by Trans(X) the set of all translations of X. It also describes Euclidean spaces and assumes that the real vector space X is of finite dimension n and that d is a Euclidean metric on X. Finally, it discusses Euclidean representations and the standard metric.

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