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Combining several results on related (or conjugate) connections, defined on banachable fibre bundles, we set up a machinery, which permits to study various transformations of linear connections. Global and local methods are applied throughout. As an application, we get an extension of the classical affine transformations to the context of infinite-dimensional vector bundles. Another application shows that, realising the ordinary linear differential equations (in Banach spaces) as connections, we get the usual transformations of (equivalent) equations. Thus, some classical results on differential equations, such as the Theorem of Floquet, can have a “geometric” interpretation.

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Abstract  

Pseudoconnections (or quasi connections) were defined as a generalization of linear connections by Y.-C. Wong in [14], and were developed mainly by Italian and Rumanian mathematicians. The purpose of this paper is to study some properties of a special type of pseudoconnections: the so-called associated pseudoconnections oginirating from linear connections in a very simple manner. In 1 we give a necessary and sufficient condition for a pseudoconnection to be associated, the in 2 we study the geodesics of an associated pseudoconnection. This notion has an immediate application in Finsler geometry, this is the theme of 3. Some questions connecting the curvature of associated pseudoconnections were studied by the author in [7].

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The purpose of this paper is to study the principal fibre bundle (P, M, G, π p ) with Lie group G, where M admits Lorentzian almost paracontact structure (Ø, ξ p , η p , g) satisfying certain condtions on (1, 1) tensor field J, indeed possesses an almost product structure on the principal fibre bundle. In the later sections, we have defined trilinear frame bundle and have proved that the trilinear frame bundle is the principal bundle and have proved in Theorem 5.1 that the Jacobian map π * is the isomorphism.

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The authors' hypothesis is that the members of the tricarboxylic acid cycle (TCA cycle) such as citrate decrease in association with increased ketone body formation. To prove this hypothesis the connection between ketone bodies and citrate formation of milk was studied. A fluorimetric method was used to determine citrate and a headspace sampling gas chromatographic (GC) method was developed for determination of ketone bodies. Under real conditions of milk sampling, transport and storage, preserved milk samples of 119 clinically healthy dairy cows obtained in the 48 hours after milking were investigated. A low level of acetoacetate (ACAC) was found in all samples. This fact can be explained by the spontaneous decarboxylation of acetoacetate during sample storage (previously decarboxylised acetoacetate = pdACAC) and, consequently, the majority of the amount of acetoacetate in the samples (AC+pdACAC) appeared in the measured acetone concentrations. Based on the measured acetone concentration of milk samples two groups were formed retrospectively: HA (high-acetone) group (n = 41) with an AC+pdACAC concentration of ≯ 0.4 mmol/l and a LA (low-acetone) group (n = 78) with an AC+pdACAC level of ≤ 0.4 mmol/l. In the milk of cows of Group HA a positive correlation (r = +0.623) and linear connection between acetone (AC+pdACAC) and β-hydroxybutyrate (BOHB) levels was found [BOHB = 2.491 + 0.586 × (pdAC + ACAC)]. Furthermore, in this group a negative correlation between citrate and BOHB and AC+pdACAC was also established (r = -0.579). Focusing on the results of this group the authors found a significant drop of AC+pdACAC and citrate during the metabolically critical first 1-4 weeks of lactation. For this reason they suggest that simple, easy, automated methods (i.e. flow injection analysis, Fourier transformation infrared analysis) should be introduced for the simultaneous determination of acetone and citrate concentration in milk to make the evaluation of the energy status of high-producing dairy cows easier and more certain.

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. 1975 On semi-symmetric and quarter-symmetric linear connections Tensor (N.S.) 29 249 – 254 . [6] Hayden , H. A

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Trans. Amer. Math. Soc. 99 325 – 341 10.1090/S0002-9947-1961-0121751-2 . [18] Wong , Y.-C. 1962 Linear connexions with zero

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Studia Scientiarum Mathematicarum Hungarica
Authors:
Jaime Muñoz Masqué
,
María Eugenia Rosado María
, and
Ignacio Sánchez Rodríguez

., 1966 . [32] Ch.-L . Terng . Natural vector bundles and natural differential operators . Amer. J. Math ., 100 : 775 – 828 , 1978 . [33] K . Yano and A. J . Ledger . Linear connections on tangent bundles . J. London Math. Soc ., 39 : 495 – 500

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