Structure of symmetric tensors of type and tensors of type on the tangent bundle

Authors:
Kam Ping Mok, E. M. Patterson and Yung Chow Wong

Journal:
Trans. Amer. Math. Soc. **234** (1977), 253-278

MSC:
Primary 53C05; Secondary 53C15

MathSciNet review:
0500673

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Abstract: The concepts of *M*-tensor and *M*-connection on the tangent bundle *TM* of a smooth manifold *M* are used in a study of symmetric tensors of type (0, 2) and tensors of type (1, 1) on *TM*. The constructions make use of certain local frames adapted to an *M*-connection. They involve extending known results on *TM* using tensors on *M* to cases in which these tensors are replaced by *M*-tensors. Particular attention is devoted to (pseudo-) Riemannian metrics on *TM*, notably those for which the vertical distribution on *TM* is null or nonnull, and to the construction of almost product and almost complex structures on *TM*.

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DOI:
https://doi.org/10.1090/S0002-9947-1977-0500673-0

Keywords:
Tangent bundle,
Sasaki metric,
*M*-tensor,
*M*-connection,
adapted frame,
(pseudo-) Riemannian metric,
almost product structure,
almost complex structure

Article copyright:
© Copyright 1977
American Mathematical Society