A General Differential Geometry with Two Types of Linear Connection

Citation

Wyman, Max (1940) A General Differential Geometry with Two Types of Linear Connection. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/92JW-1918. https://resolver.caltech.edu/CaltechETD:etd-08262008-145426

Abstract

The object of this thesis was the study of a differential geometry for a Hausdorff space endowed with an affine linear connection and a non-holonomic linear connection. The coordinate spaces were taken to be Banach spaces. In Chapter II we define the notion of a non-holonomic contravariant vector field, and by means of the non-holonomic linear connection introduce the operation of covariant differentiation. It was then found that many of the formal tensor theorems carried over to such spaces. For certain types of Hausdorff space it is possible to develop a normal representation theory, and by means of it to obtain normal non-holonomic vector forms. This then enables us to generalize the Michal-Hyers replacement theorem for differential invariants. Chapter IV is concerned with the determination of nonholonomic linear connections. This leads to the consideration of interspace adjoints for linear functions. In the main the results obtained in this thesis are generalizations of results obtained for finite dimensional spaces by A.D. Michal and J. L. Botsford. However the projective theory developed in Chapter V is new for spaces of finite dimension.

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