Affine differential geometry and holomorphic curves

1990 ◽  
pp. 15-21 ◽  
Author(s):  
Eugenio Calabi
Keyword(s):  
Differential Geometry ◽  
Holomorphic Curves ◽  
Affine Differential Geometry

scholarly journals A Note on Pseudo-Umbilical Submanifolds of Hessian Manifolds with Constant Hessian Sectional Curvature

ISRN Geometry ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-12
Author(s):  
Münevver Yildirim Yilmaz ◽  
Mehmet Bektaş
Keyword(s):  
Differential Geometry ◽  
Sectional Curvature ◽  
Affine Differential Geometry

The geometry of Hessian manifold, as a branch of statistics, physics, Kaehlerian, and affine differential geometry, is deeply fruitful and a new field for scientists. However, inspite of its importance submanifolds and curvature conditions of it have not been so well known yet. In this paper, we focus on the pseudo-umbilical submanifolds on Hessian manifold with constant Hessian sectional curvature and using sectional curvature conditions we obtain new results on it.

scholarly journals On the Affine Image of a Rational Surface of Revolution

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2061
Author(s):  
Juan G. Alcázar
Keyword(s):  
Differential Geometry ◽  
Affine Transformation ◽  
Rational Surface ◽  
Affine Differential Geometry ◽  
Affine Mapping ◽  
Surfaces Of Revolution ◽  
Surface Of Revolution ◽  
Affine Image ◽  
Fixed Line ◽  
The Given

We study the properties of the image of a rational surface of revolution under a nonsingular affine mapping. We prove that this image has a notable property, namely that all the affine normal lines, a concept that appears in the context of affine differential geometry, created by Blaschke in the first decades of the 20th century, intersect a fixed line. Given a rational surface with this property, which can be algorithmically checked, we provide an algorithmic method to find a surface of revolution, if it exists, whose image under an affine mapping is the given surface; the algorithm also finds the affine transformation mapping one surface onto the other. Finally, we also prove that the only rational affine surfaces of rotation, a generalization of surfaces of revolution that arises in the context of affine differential geometry, and which includes surfaces of revolution as a subtype, affinely transforming into a surface of revolution are the surfaces of revolution, and that in that case the affine mapping must be a similarity.

The Bountiful Intersection of Differential Geometry, Mechanics, and Control Theory

2018 ◽  
Vol 1 (1) ◽  
pp. 135-158 ◽  
Author(s):  
Andrew D. Lewis
Keyword(s):  
Differential Geometry ◽  
Control Theory ◽  
Crucial Role ◽  
Affine Differential Geometry ◽  
Geometric Methods ◽  
History Of ◽  
And Control

The areas of mechanics and control theory have a rich and productive history of interaction with the broad mathematical subject of differential geometry. This article provides an overview of these sorts of interplay in the areas of Riemannian and affine differential geometry and the geometry of vector distributions. It emphasizes areas where differential geometric methods have played a crucial role in solving problems whose solutions are difficult to achieve without access to these methods. It also emphasizes a concise and elegant presentation of the approach, rather than a detailed and concrete presentation. The results overviewed, while forming a coherent and elegant body of work, are limited in scope. The review closes with a discussion of why the approach is limited and a brief consideration of issues that must be resolved before the results of the type presented here can be extended.

Riemann extensions and affine differential geometry

1988 ◽  
Vol 13 (3-4) ◽  
pp. 403-408 ◽  
Author(s):  
Tom Willmore
Keyword(s):  
Differential Geometry ◽  
Affine Differential Geometry

scholarly journals Generic affine differential geometry of plane curves

1998 ◽  
Vol 41 (2) ◽  
pp. 315-324 ◽  
Author(s):  
Shyuichi Izumiya ◽  
Takasi Sano
Keyword(s):  
Differential Geometry ◽  
Singularity Theory ◽  
Affine Differential Geometry ◽  
Plane Curves ◽  
View Point ◽  
Smooth Functions ◽  
Affine Invariants

We study affine invariants of plane curves from the view point of the singularity theory of smooth functions

scholarly journals Gheorghe Ţiţeica and the origins of affine differential geometry

2009 ◽  
Vol 36 (2) ◽  
pp. 161-170 ◽  
Author(s):  
Alfonso F. Agnew ◽  
Alexandru Bobe ◽  
Wladimir G. Boskoff ◽  
Bogdan D. Suceavă
Keyword(s):  
Differential Geometry ◽  
Affine Differential Geometry

scholarly journals Generic affine differential geometry of space curves

1998 ◽  
Vol 128 (2) ◽  
pp. 301-314 ◽  
Author(s):  
Shyuichi Izumiya ◽  
Takasi Sano
Keyword(s):  
Differential Geometry ◽  
Singularity Theory ◽  
Affine Differential Geometry ◽  
Smooth Functions ◽  
Space Curves ◽  
Affine Invariants

We study affine invariants of space curves from the viewpoint of singularity theory of smooth functions. With the aid of singularity theory, we define a new equi-affine frame for space curves. We also introduce two surfaces associated with this equi-affine frame and give a generic classification of the singularities of those surfaces.

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