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A CLIFFORD-TYPE STRUCTURE

A CLIFFORD-TYPE STRUCTURE DEMONSTRATIO MATHEMATICAVol. XXXVINo 32003Wiestaw KrôlikowskiA CLIFFORD-TYPE STRUCTUREIntroductionIn the paper a Clifford-type structure is introduced and some considerations on Clifford-type manifolds are developped. First of all an analog of thefundamental 2-form of complex analysis is defined and using it a decomposition analogous to the Hodge Decomposition Theorem for Kahler manifoldsis given for Clifford-type manifolds. By the Chern Theorem [5] we get anincreasing sequence of Betti numbers for Clifford-type manifolds.1. A Clifford-type structureLet V be a real vector space.DEFINITION 1.1. An almost Clifford-type structure Cn on V is a set of nalmost complex structures { i i , . . . , /„} such thatIaIp+ Ipla= —26apld,a, (3 = 1,...,n,where Id stands for the identity endomorphism of V, 6 denotes the „Kronecker delta".REMARK 1.1. a) If n = 1, then Ci = {/} with I2 = -Id. Thus, C\ is nothingbut an almost complex structure. Recall that the standard form of an almostcomplex structure looks as follows:/ O =( - ° /O)'{ I =ID)provided that V has an even dimension (see, e.g. [10]).b) If n = 2, then C2 = {I, J} with I2 = J2 = -Id and IJ + JI = 0.Define K := I J, then IJK = http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

A CLIFFORD-TYPE STRUCTURE

Demonstratio Mathematica , Volume 36 (3): 16 – Jul 1, 2003

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Publisher
de Gruyter
Copyright
© by Wiesław Królikowski
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-2003-0309
Publisher site
See Article on Publisher Site

Abstract

DEMONSTRATIO MATHEMATICAVol. XXXVINo 32003Wiestaw KrôlikowskiA CLIFFORD-TYPE STRUCTUREIntroductionIn the paper a Clifford-type structure is introduced and some considerations on Clifford-type manifolds are developped. First of all an analog of thefundamental 2-form of complex analysis is defined and using it a decomposition analogous to the Hodge Decomposition Theorem for Kahler manifoldsis given for Clifford-type manifolds. By the Chern Theorem [5] we get anincreasing sequence of Betti numbers for Clifford-type manifolds.1. A Clifford-type structureLet V be a real vector space.DEFINITION 1.1. An almost Clifford-type structure Cn on V is a set of nalmost complex structures { i i , . . . , /„} such thatIaIp+ Ipla= —26apld,a, (3 = 1,...,n,where Id stands for the identity endomorphism of V, 6 denotes the „Kronecker delta".REMARK 1.1. a) If n = 1, then Ci = {/} with I2 = -Id. Thus, C\ is nothingbut an almost complex structure. Recall that the standard form of an almostcomplex structure looks as follows:/ O =( - ° /O)'{ I =ID)provided that V has an even dimension (see, e.g. [10]).b) If n = 2, then C2 = {I, J} with I2 = J2 = -Id and IJ + JI = 0.Define K := I J, then IJK =

Journal

Demonstratio Mathematicade Gruyter

Published: Jul 1, 2003

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