Abstract

We discuss the subspaces of an almost šœ‘-Lagrange space (APL space in short). We obtain the induced nonlinear connection, coefficients of coupling, coefficients of induced tangent and induced normal connections, the Gauss-Weingarten formulae, and the Gauss-Codazzi equations for a subspace of an APL-space. Some consequences of the Gauss-Weingarten formulae have also been discussed.

1. Introduction

The credit for introducing the geometry of Lagrange spaces and their subspaces goes to the famous Romanian geometer Miron [1]. He developed the theory of subspaces of a Lagrange space together with Bejancu [2]. Miron and Anastasiei [3] and Sakaguchi [4] studied the subspaces of generalized Lagrange spaces (GL spaces in short). Antonelli and Hrimiuc [5, 6] introduced the concept of šœ‘-Lagrangians and studied šœ‘-Lagrange manifolds. Generalizing the notion of a šœ‘-Lagrange manifold, the present authors recently studied the geometry of an almost šœ‘-Lagrange space (APL space briefly) and obtained the fundamental entities related to such space [7]. This paper is devoted to the subspaces of an APL space.

Let š¹š‘›=(š‘€,š¹(š‘„,š‘¦)) be an š‘›-dimensional Finsler space and šœ‘āˆ¶ā„+ā†’ā„ a smooth function. If the function šœ‘ has the following properties:(a)šœ‘ī…ž(š‘”)ā‰ 0, (b)šœ‘ī…ž(š‘”)+šœ‘ī…žī…ž(š‘”)ā‰ 0, for every š‘”āˆˆIm(š¹2),then the Lagrangian given by ī€·š¹šæ(š‘„,š‘¦)=šœ‘2ī€ø+š“š‘–(š‘„)š‘¦š‘–+š‘ˆ(š‘„),(1.1) where š“š‘–(š‘„) is a covector and š‘ˆ(š‘„) is a smooth function, is a regular Lagrangian [7]. The space šæš‘›=(š‘€,šæ(š‘„,š‘¦)) is a Lagrange space. The present authors [7] called such space as an almost šœ‘-Lagrange space (shortly APL space) associated to the Finsler space š¹š‘›. An APL space reduces to a šœ‘-Lagrange space if and only if š“š‘–(š‘„)=0 and š‘ˆ(š‘„)=0. We take š‘”š‘–š‘—=12Ģ‡šœ•š‘–Ģ‡šœ•š‘—š¹2,š‘Žš‘–š‘—=12Ģ‡šœ•š‘–Ģ‡šœ•š‘—šæ,whereĢ‡šœ•š‘–ā‰”šœ•šœ•š‘¦š‘–.(1.2) We indicate all the geometrical objects related to š¹š‘› by putting a small circle ā€œāˆ˜ā€ over them. Equations (1.2), in view of (1.1), provide the following expressions for š‘Žš‘–š‘— and its inverse (cf. [7]): š‘Žš‘–š‘—=šœ‘ī…žā‹…ī‚µš‘”š‘–š‘—+2šœ‘ī…žī…žšœ‘ī…žāˆ˜š‘¦š‘–āˆ˜š‘¦š‘—ī‚¶,š‘Žš‘–š‘—=1ī‚µš‘”šœ‘ā€²š‘–š‘—āˆ’2šœ‘ī…žī…žšœ‘ā€²+2š¹2šœ‘ī…žī…žš‘¦š‘–š‘¦š‘—ī‚¶,(1.3) where š‘”š‘–š‘—š‘¦š‘—=āˆ˜š‘¦š‘–.

Let ī‚š‘€ be a smooth manifold of dimension š‘š, 1<š‘š<š‘›, immersed in š‘€ by immersion ī‚š‘–āˆ¶š‘€ā†’š‘€. The immersion š‘– induces an immersion š‘‡š‘–ī‚āˆ¶š‘‡š‘€ā†’š‘‡š‘€ making the following diagram commutative: š‘‡ī‚š‘€š‘‡š‘–ī‚š‘€āŸ¶š‘–āŸ¶š‘‡š‘€ī„šœ‹ā†“ā†“šœ‹š‘€.(1.4)

Let (š‘¢š›¼,š‘£š›¼) (throughout the paper, the Greek indices š›¼,š›½,š›¾,ā€¦ run from 1 to š‘š) be local coordinates on š‘‡ī‚š‘€. The restriction of the Lagrangian šæ on š‘‡ī‚š‘€ is šæ(š‘¢,š‘£)=šæ(š‘„(š‘¢),š‘¦(š‘¢,š‘£)). Let š‘Žš›¼š›½=(1/2)(šœ•2ī„šæ/šœ•š‘¢š›¼šœ•š‘¢š›½). Then, we have (cf. [8]) š‘Žš›¼š›½=šµš‘–š›¼šµš‘—š›½š‘Žš‘–š‘— where šµš‘–š›¼(š‘¢)=šœ•š‘„š‘–/šœ•š‘¢š›¼ are the projection factors. The pair ī‚šæš‘šī‚ī„=(š‘€,šæ(š‘¢,š‘£)) is also a Lagrange space, called the subspace of šæš‘›. For the natural bases (šœ•/šœ•š‘„š‘–,šœ•/šœ•š‘¦š‘–) on TM and (šœ•/šœ•š‘¢š›¼,šœ•/šœ•š‘£š›¼) on š‘‡ī‚š‘€, we have [8] šœ•šœ•š‘¢š›¼=šµš‘–š›¼šœ•šœ•š‘„š‘–+šµš‘–0š›¼šœ•šœ•š‘¦š‘–,šœ•šœ•š‘£š›¼=šµš‘–š›¼šœ•šœ•š‘¦š‘–,(1.5) where šµš‘–0š›¼=šµš‘–š›½š›¼š‘£š›½,šµš‘–š›½š›¼=šœ•2š‘„š‘–/šœ•š‘¢š›¼šœ•š‘¢š›½.

For the bases (š‘‘š‘„š‘–,š‘‘š‘¦š‘–) and (š‘‘š‘¢š›¼,š‘‘š‘£š›¼), we have š‘‘š‘„š‘–=šµš‘–š›¼š‘‘š‘¢š›¼,š‘‘š‘¦š‘–=šµš‘–š›¼š‘‘š‘£š›¼+šµš‘–0š›¼š‘‘š‘¢š›¼.(1.6) Since (šµš‘–š›¼) are š‘š linearly independent vector fields tangent to ī‚š‘€, a vector field šœ‰š‘–(š‘„,š‘¦) is normal to ī‚š‘€ along š‘‡ī‚š‘€ if on š‘‡ī‚š‘€, we have š‘Žš‘–š‘—(š‘„(š‘¢),š‘¦(š‘¢,š‘£))šµš‘–š›¼šœ‰š‘—=0,āˆ€š›¼=1,2,ā€¦,š‘š.(1.7) There are, at least locally, (š‘›āˆ’š‘š) unit vector fields šµš‘–š‘Ž(š‘¢,š‘£)(š‘Ž=š‘š+1,š‘š+2,ā€¦,š‘›) normal to ī‚š‘€ and mutually orthonormal, that is, š‘Žš‘–š‘—šµš‘–š›¼šµš‘—š‘=0,š‘Žš‘–š‘—šµš‘–š‘Žšµš‘—š‘=š›æš‘Žš‘,(š‘Ž,š‘=š‘š+1,š‘š+2,ā€¦,š‘›).(1.8) Thus, at every point ī‚š‘€(š‘¢,š‘£)āˆˆš‘‡, we have a moving frame ā„œ=((š‘¢,š‘£),šµš‘–š›¼(š‘¢,š‘£),šµš‘–š‘Ž(š‘¢,š‘£)). Using (1.3) in the first expression of (1.8) and keeping āˆ˜š‘¦š‘–šµš‘–š‘Ž=0 (this fact is clear from š‘”š‘–š‘—š‘¦š‘–šµš‘—š‘Ž=0) in view, we observe that šµš‘–š‘Ž's are normal to ī‚š‘€ with respect to šæš‘› if and only if they are so with respect to š¹š‘›. The dual frame of ā„œ is ā„œāˆ—=((š‘¢,š‘£),šµš›¼š‘–(š‘¢,š‘£),šµš‘Žš‘–(š‘¢,š‘£)) with the following duality conditions: šµš‘–š›¼šµš›½š‘–=š›æš›½š›¼,šµš‘–š‘Žšµš›½š‘–=0,šµš‘–š›¼šµš‘š‘–=0,šµš‘–š‘Žšµš‘š‘–=š›æš‘š‘Ž,šµš‘–š‘Žšµš‘Žš‘—+šµš‘–š›¼šµš›¼š‘—=š›æš‘–š‘—.(1.9) We will make use of the following results due to the present authors [7], during further discussion.

Theorem 1.1 1.1 (cf. [7]). The canonical nonlinear connection of an APL space šæš‘› has the local coefficients given by š‘š‘–š‘—=āˆ˜š‘š‘–š‘—āˆ’š‘‰š‘–š‘—,(1.10) where š‘‰š‘–š‘—=(1/2)š¹š‘–š‘—āˆ’š‘†š‘—š‘–š‘Ÿ(2š¹š‘Ÿš‘˜š‘¦š‘˜+šœ•š‘Ÿš‘ˆ), š‘†š‘—š‘–š‘Ÿ=12šœ‘ā€²āˆ˜š¶š‘–š‘žš‘—š‘”š‘žš‘Ÿ+12šœ‘ī…žī…žšœ‘ā€²2š‘”āˆ˜š‘–š‘Ÿš‘¦š‘—+šœ‘ī…žī…žī€·š›æš‘Ÿš‘—š‘¦š‘–+š›æš‘–š‘—š‘¦š‘Ÿī€ø2šœ‘ī…žī€·šœ‘ī…ž+2š¹2šœ‘ī…žī…žī€ø+šœ‘ā€²2šœ‘ī…žī…žī…žāˆ’2šœ‘ī…žī…ž3š¹2āˆ’4šœ‘ā€²šœ‘ī…žī…ž22šœ‘ā€²2ī€·šœ‘ī…ž+2š¹2šœ‘ī…žī…žī€ø2š‘¦š‘–āˆ˜š‘¦š‘—š‘¦š‘Ÿ,š¹š‘Ÿš‘˜(1š‘„)=2ī€·šœ•š‘Ÿš“š‘˜āˆ’šœ•š‘˜š“š‘Ÿī€ø,š¹š‘–š‘—=š‘Žš‘–š‘˜š¹š‘˜š‘—.(1.11)

Theorem 1.2 (cf. [7]). The coefficients of the canonical metrical d-connection š¶Ī“(š‘) of an APL space šæš‘› are given by š¶š‘–š‘—š‘˜=āˆ˜š¶š‘–š‘—š‘˜+šœ‘ī…žī…žšœ‘ī…žī‚€š›æš‘–š‘—āˆ˜š‘¦š‘˜+š›æš‘–š‘˜āˆ˜š‘¦š‘—ī‚+šœ‘ī…žī…žšœ‘ā€²+2š¹2šœ‘ī…žī…žš‘”š‘—š‘˜š‘¦š‘–+2ī€·šœ‘ī…žī…žī…žšœ‘ā€²āˆ’2šœ‘ī…žī…ž2ī€øšœ‘ī…žī€·šœ‘ī…ž+2š¹2šœ‘ī…žī…žī€øš‘¦š‘–āˆ˜š‘¦š‘—āˆ˜š‘¦š‘˜,(1.12)šæš‘–š‘—š‘˜=āˆ˜šæš‘–š‘—š‘˜+š‘‰š‘Ÿš‘˜š¶š‘–š‘—š‘Ÿ+š‘‰š‘Ÿš‘—š¶š‘–š‘˜š‘Ÿ+š‘‰š‘Ÿš‘š‘Žš‘–š‘š¶š‘Ÿš‘˜š‘—.(1.13)

For basic notations related to a Finsler space, a Lagrange space, and their subspaces, we refer to the books [8, 9].

2. Induced Nonlinear Connection

Let ī„ī‚š‘š‘=(š›¼š›½(š‘¢,š‘£)) be a nonlinear connection for ī‚šæš‘šī‚ī„=(š‘€,šæ(š‘¢,š‘£)). The adapted basis of š‘‡(š‘¢,š‘£)š‘‡ī‚š‘€ induced by ī„š‘ is (š›æ/š›æš‘¢š›¼=š›æš›¼,šœ•/šœ•š‘£š›¼=Ģ‡šœ•š›¼), where š›æš›¼=šœ•š›¼āˆ’ī‚š‘š›½š›¼Ģ‡šœ•š›½.(2.1) The dual basis (cobasis) of the adapted basis (š›æš›¼,Ģ‡šœ•š›¼) is (š‘‘š‘¢š›¼,š›æš‘£š›¼=š‘‘š‘£š›¼+ī‚š‘š›¼š›½š‘‘š‘¢š›½).

Definition 2.1 (cf. [8]). A nonlinear connection ī„ī‚š‘š‘=(š›¼š›½(š‘¢,š‘£)) of ī‚šæš‘š is said to be induced by the canonical nonlinear connection š‘ if the following equation holds good: š›æš‘£š›¼=šµš›¼š‘–š›æš‘¦š‘–.(2.2) The local coefficients of the induced nonlinear connection ī„ī‚š‘š‘=(š›¼š›½(š‘¢,š‘£)) for the subspace ī‚šæš‘šī‚ī„=(š‘€,šæ(š‘¢,š‘£)) of a Lagrange space šæš‘›=(š‘€,šæ(š‘„,š‘¦)) are given by (cf. [8]) ī‚š‘š›¼š›½=šµš›¼š‘–ī‚€š‘š‘–š‘—šµš‘—š›½+šµš‘–0š›½ī‚,(2.3)š‘š‘–š‘— being the local coefficients of canonical nonlinear connection š‘ of the Lagrange space šæš‘›=(š‘€,šæ(š‘„,š‘¦)). Now using (1.10) in (2.3), we get ī‚š‘š›¼š›½=šµš›¼š‘–ī‚µāˆ˜š‘š‘–š‘—šµš‘—š›½+šµš‘–0š›½ī‚¶āˆ’šµš›¼š‘–š‘‰š‘–š‘—šµš‘—š›½.(2.4) If we take āˆ˜ī„š‘š›¼š›½=šµš›¼š‘–(āˆ˜š‘š‘–š‘—šµš‘—š›½+šµš‘–0š›½), it follows from (2.4) that ī‚š‘š›¼š›½=āˆ˜ī„š‘š›¼š›½āˆ’šµš›¼š‘–š‘‰š‘–š‘—šµš‘—š›½.(2.5) Thus, we have the following.

Theorem 2.2. The local coefficients of the induced nonlinear connection ī„š‘ of the subspace ī‚šæš‘š of an APL space šæš‘› are given by (2.5).

In view of (2.5), (2.1) takes the following form, for the subspace ī‚šæš‘š of an APL space šæš‘›: š›æš›½=āˆ˜š›æš›½+šµš›¼š‘š‘‰š‘š‘—šµš‘—š›½Ģ‡šœ•š›¼,(2.6) where āˆ˜š›æš›½=šœ•š›½āˆ’āˆ˜ī„š‘š›¼š›½Ģ‡šœ•š›¼.

We can put (š‘‘š‘„š‘–,š›æš‘¦š‘–) as (cf. [8]) š‘‘š‘„š‘–=šµš‘–š›¼š‘‘š‘¢š›¼,š›æš‘¦š‘–=šµš‘–š›¼š›æš‘¦š›¼+šµš‘–š‘Žš»š‘Žš›¼š‘‘š‘¢š›¼,(2.7) where š»š‘Žš›¼=šµš‘Žš‘–ī‚€š‘š‘–š‘—šµš‘—š›¼+šµš‘–0š›¼ī‚.(2.8) Using (1.10) in (2.8) and simplifying, we get š»š‘Žš›¼=šµš‘Žš‘–ī‚µāˆ˜š‘š‘–š‘—šµš‘—š›¼+šµš‘–0š›¼ī‚¶āˆ’šµš‘Žš‘–š‘‰š‘–š‘—šµš‘—š›¼.(2.9) Taking āˆ˜š»š‘Žš›¼=šµš‘Žš‘–(āˆ˜š‘š‘–š‘—šµš‘—š›¼+šµš‘–0š›¼), in (2.9), it follows that š»š‘Žš›¼=āˆ˜š»š‘Žš›¼āˆ’šµš‘Žš‘–š‘‰š‘–š‘—šµš‘—š›¼.(2.10) Now, š‘‘š‘„š‘–=šµš‘–š›¼š‘‘š‘¢š›¼,š›æš‘¦š‘–=šµš‘–š›¼š›æš‘¦š›¼ if and only if š»š‘Žš›¼=0, that is, if and only if āˆ˜š»š‘Žš›¼=šµš‘Žš‘–š‘‰š‘–š‘—šµš‘—š›¼. Thus, we have the following.

Theorem 2.3. The adapted cobasis (š‘‘š‘„š‘–,š›æš‘¦š‘–) of the basis (šœ•/šœ•š‘„š‘–,šœ•/šœ•š‘¦š‘–) induced by the nonlinear connection š‘ of an APL space šæš‘› is of the form š‘‘š‘„š‘–=šµš‘–š›¼š‘‘š‘¢š›¼,š›æš‘¦š‘–=šµš‘–š›¼š›æš‘¦š›¼ if and only if āˆ˜š»š‘Žš›¼=šµš‘Žš‘–š‘‰š‘–š‘—šµš‘—š›¼.

Definition 2.4 (cf. [8]). Let š·=š·Ī“(š‘) be the canonical metrical š‘‘-connection of šæš‘›. An operator ī„š· is said to be a coupling of š· with ī„š‘ if ī„š·š‘‹š‘–=š‘‹š‘–|š›¼š‘‘š‘¢š›¼+š‘‹š‘–|š›¼š›æš‘£š›¼,(2.11) where š‘‹š‘–|š›¼=š›æš›¼š‘‹š‘–+š‘‹š‘—ī‚šæš‘–š‘—š›¼,š‘‹š‘–|š›¼=Ģ‡šœ•š›¼š‘‹š‘–+š‘‹š‘—ī‚š¶š‘–š‘—š›¼.
The coefficients (ī‚šæš‘–š‘—š›¼,ī‚š¶š‘–š‘—š›¼) of coupling ī„š· of š· with ī„š‘ are given by ī‚šæš‘–š‘—š›¼=šæš‘–š‘—š‘˜šµš‘˜š›¼+š¶š‘–š‘—š‘˜šµš‘˜š‘Žš»š‘Žš›¼,(2.12)ī‚š¶š‘–š‘—š›¼=š¶š‘–š‘—š‘˜šµš‘˜š›¼.(2.13) Using (1.12) and (1.13) in (2.12), we have ī‚šæš‘–š‘—š›½=ī‚µāˆ˜šæš‘–š‘—š‘˜+š‘‰š‘Ÿš‘˜š¶š‘–š‘—š‘Ÿ+š‘‰š‘Ÿš‘—š¶š‘–š‘˜š‘Ÿ+š‘‰š‘Ÿš‘š‘Žš‘–š‘š¶š‘Ÿš‘˜š‘—ī‚¶šµš‘˜š›½+ī‚øāˆ˜š¶š‘–š‘—š‘˜+šœ‘ī…žī…žšœ‘ī…žī‚€š›æš‘–š‘—āˆ˜š‘¦š‘˜+š›æš‘–š‘˜āˆ˜š‘¦š‘—ī‚+šœ‘ī…žī…žšœ‘ā€²+2š¹2šœ‘ī…žī…žš‘”š‘—š‘˜š‘¦š‘–+2ī€·šœ‘ī…žī…žī…žšœ‘ī…žāˆ’2šœ‘ī…žī…ž2ī€øšœ‘ī…žī€·šœ‘ī…ž+2š¹2šœ‘ī…žī…žī€øš‘¦š‘–āˆ˜š‘¦š‘—āˆ˜š‘¦š‘˜īƒ­šµš‘˜š‘Žš»š‘Žš›½.(2.14) In view of (2.10) and āˆ˜š‘¦š‘–šµš‘–š‘Ž=0, (2.14) becomes ī‚šæš‘–š‘—š›½=ī‚µāˆ˜šæš‘–š‘—š‘˜šµš‘˜š›½+āˆ˜š¶š‘–š‘—š‘˜šµš‘˜š‘Žāˆ˜š»š‘Žš›½ī‚¶+ī‚µš‘‰š‘Ÿš‘˜š¶š‘–š‘—š‘Ÿ+š‘‰š‘Ÿš‘—š¶š‘–š‘˜š‘Ÿ+š‘‰š‘Ÿš‘š‘Žš‘–š‘š¶š‘Ÿš‘˜š‘—āˆ’āˆ˜š¶š‘–š‘—š‘Ÿšµš‘Ÿš‘šµš‘š‘š‘‰š‘š‘˜ī‚¶šµš‘˜š›½+ī‚µšœ‘ī…žī…žšœ‘ā€²āˆ˜š‘¦š‘—š›æš‘–š‘˜+šœ‘ī…žī…žšœ‘ā€²+2š¹2šœ‘ī…žī…žš‘”š‘—š‘˜š‘¦š‘–ī‚¶šµš‘˜š‘Žš»š‘Žš›½,(2.15) that is, ī‚šæš‘–š‘—š›½=āˆ˜ī„šæš‘–š‘—š›½+ī‚µš‘‰š‘Ÿš‘˜š¶š‘–š‘—š‘Ÿ+š‘‰š‘Ÿš‘—š¶š‘–š‘˜š‘Ÿ+š‘‰š‘Ÿš‘š‘Žš‘–š‘š¶š‘Ÿš‘˜š‘—āˆ’āˆ˜š¶š‘–š‘—š‘Ÿšµš‘Ÿš‘šµš‘š‘š‘‰š‘š‘˜ī‚¶šµš‘˜š›½+ī‚µšœ‘ī…žī…žšœ‘ā€²āˆ˜š‘¦š‘—š›æš‘–š‘˜+šœ‘ī…žī…žšœ‘ā€²+2š¹2šœ‘ī…žī…žš‘”š‘—š‘˜š‘¦š‘–ī‚¶šµš‘˜š‘Žš»š‘Žš›½,(2.16) where āˆ˜ī„šæš‘–š‘—š›½=āˆ˜šæš‘–š‘—š‘˜šµš‘˜š›½+āˆ˜š¶š‘–š‘—š‘˜šµš‘˜š‘Žāˆ˜š»š‘Žš›½.
Using (1.12) in (2.13), we find that ī‚š¶š‘–š‘—š›½=āˆ˜š¶š‘–š‘—š‘˜šµš‘˜š›½+īƒ©šœ‘ī…žī…žī‚€š›æšœ‘ā€²š‘–š‘—āˆ˜š‘¦š‘˜+š›æš‘–š‘˜āˆ˜š‘¦š‘—ī‚+šœ‘ī…žī…žšœ‘ā€²+2š¹2šœ‘ī…žī…žš‘”š‘—š‘˜š‘¦š‘–+2ī€·šœ‘ī…žī…žī…žšœ‘ā€²āˆ’2šœ‘ī…žī…ž2ī€øī€·šœ‘ā€²šœ‘ā€²+2š¹2šœ‘ī…žī…žī€øš‘¦š‘–āˆ˜š‘¦š‘—āˆ˜š‘¦š‘˜īƒŖšµš‘˜š›½,(2.17) that is, ī‚š¶š‘–š‘—š›½=āˆ˜ī„š¶š‘–š‘—š›½+īƒ©šœ‘ī…žī…žī‚€š›æšœ‘ā€²š‘–š‘—āˆ˜š‘¦š‘˜+š›æš‘–š‘˜āˆ˜š‘¦š‘—ī‚+šœ‘ī…žī…žšœ‘ā€²+2š¹2šœ‘ī…žī…žš‘”š‘—š‘˜š‘¦š‘–+2ī€·šœ‘ī…žī…žī…žšœ‘ā€²āˆ’2šœ‘ī…žī…ž2ī€øī€·šœ‘ā€²šœ‘ā€²+2š¹2šœ‘ī…žī…žī€øš‘¦š‘–āˆ˜š‘¦š‘—āˆ˜š‘¦š‘˜īƒŖšµš‘˜š›½,(2.18) where āˆ˜ī„š¶š‘–š‘—š›½=āˆ˜š¶š‘–š‘—š‘˜šµš‘˜š›½. Thus, we have the following.

Theorem 2.5. The coefficients of coupling for the subspace ī‚šæš‘š of an APL space šæš‘› are given by (2.16) and (2.18).

Definition 2.6 (cf. [8]). An operator š·š‘‡ given by š·š‘‡š‘‹š›¼=š‘‹š›¼|š›½š‘‘š‘¢š›½+š‘‹š›¼|š›½š›æš‘£š›½,(2.19) where š‘‹š›¼|š›½=š›æš›½š‘‹š›¼+š‘‹š›¾šæš›¼š›¾š›½,š‘‹š›¼|š›½=Ģ‡šœ•š›½š‘‹š›¼+š‘‹š›¾š¶š›¼š›¾š›½, is called the induced tangent connection by š·. This defines an š‘-linear connection for ī‚šæš‘š.
The coefficients (šæš›¼š›¾š›½,š¶š›¼š›¾š›½) of š·š‘‡ are given by šæš›¼š›½š›¾=šµš›¼š‘–ī‚€šµš‘–š›½š›¾+šµš‘—š›½ī‚šæš‘–š‘—š›¾ī‚,(2.20)š¶š›¼š›½š›¾=šµš›¼š‘–šµš‘—š›½ī‚š¶š‘–š‘—š›¾.(2.21) Using (2.16) in (2.20), we get šæš›¼š›½š›¾=šµš›¼š‘–šµš‘–š›½š›¾+šµš‘—š›½šµš›¼š‘–īƒ¬āˆ˜ī„šæš‘–š‘—š›¾+ī‚µš‘‰š‘Ÿš‘˜š¶š‘–š‘—š‘Ÿ+š‘‰š‘Ÿš‘—š¶š‘–š‘˜š‘Ÿ+š‘‰š‘Ÿš‘š‘Žš‘–š‘š¶š‘Ÿš‘˜š‘—āˆ’āˆ˜š¶š‘–š‘—š‘Ÿšµš‘Ÿš‘šµš‘š‘š‘‰š‘š‘˜ī‚¶šµš‘˜š›¾+ī‚µšœ‘ī…žī…žšœ‘ā€²āˆ˜š‘¦š‘—š›æš‘–š‘˜+šœ‘ī…žī…žšœ‘ā€²+2š¹2šœ‘ī…žī…žš‘”š‘—š‘˜š‘¦š‘–ī‚¶šµš‘˜š‘Žš»š‘Žš›¾ī‚¹,(2.22) that is, šæš›¼š›½š›¾=šµš›¼š‘–īƒ©šµš‘–š›½š›¾+āˆ˜ī„šæš‘–š‘—š›¾šµš‘—š›½īƒŖ+šµš›¼š‘–šµš‘—š›½š‘‰ī‚øī‚µš‘Ÿš‘˜š¶š‘–š‘—š‘Ÿ+š‘‰š‘Ÿš‘—š¶š‘–š‘˜š‘Ÿ+š‘‰š‘Ÿš‘š‘Žš‘–š‘š¶š‘Ÿš‘˜š‘—āˆ’āˆ˜š¶š‘–š‘—š‘Ÿšµš‘Ÿš‘šµš‘š‘š‘‰š‘š‘˜ī‚¶šµš‘˜š›¾+ī‚µšœ‘ī…žī…žšœ‘ā€²āˆ˜š‘¦š‘—š›æš‘–š‘˜+šœ‘ī…žī…žšœ‘ā€²+2š¹2šœ‘ī…žī…žš‘”š‘—š‘˜š‘¦š‘–ī‚¶šµš‘˜š‘Žš»š‘Žš›¾ī‚¹.(2.23) If we take āˆ˜šæš›¼š›½š›¾=šµš›¼š‘–(šµš‘–š›½š›¾+āˆ˜ī„šæš‘–š‘—š›¾šµš‘—š›½), the last expression gives šæš›¼š›½š›¾=āˆ˜šæš›¼š›½š›¾+šµš›¼š‘–šµš‘—š›½š‘‰ī‚øī‚µš‘Ÿš‘˜š¶š‘–š‘—š‘Ÿ+š‘‰š‘Ÿš‘—š¶š‘–š‘˜š‘Ÿ+š‘‰š‘Ÿš‘š‘Žš‘–š‘š¶š‘Ÿš‘˜š‘—āˆ’āˆ˜š¶š‘–š‘—š‘Ÿšµš‘Ÿš‘šµš‘š‘š‘‰š‘š‘˜ī‚¶šµš‘˜š›¾+ī‚µšœ‘ī…žī…žšœ‘ā€²āˆ˜š‘¦š‘—š›æš‘–š‘˜+šœ‘ī…žī…žšœ‘ā€²+2š¹2šœ‘ī…žī…žš‘”š‘—š‘˜š‘¦š‘–ī‚¶šµš‘˜š‘Žš»š‘Žš›¾ī‚¹.(2.24) Next, using (2.18) in (2.21), we obtain š¶š›¼š›½š›¾=šµš›¼š‘–šµš‘—š›½āˆ˜ī„š¶š‘–š‘—š›¾+īƒ©šœ‘ī…žī…žī‚€š›æšœ‘ā€²š‘–š‘—āˆ˜š‘¦š‘˜+š›æš‘–š‘˜āˆ˜š‘¦š‘—ī‚+šœ‘ī…žī…žšœ‘ā€²+2š¹2šœ‘ī…žī…žš‘”š‘—š‘˜š‘¦š‘–+2ī€·šœ‘ī…žī…žī…žšœ‘ā€²āˆ’2šœ‘ī…žī…ž2ī€øī€·šœ‘ā€²šœ‘ā€²+2š¹2šœ‘ī…žī…žī€øš‘¦š‘–āˆ˜š‘¦š‘—āˆ˜š‘¦š‘˜īƒŖšµš‘˜š›¾šµš›¼š‘–šµš‘—š›½.(2.25) If we take āˆ˜š¶š›¼š›½š›¾=šµš›¼š‘–šµš‘—š›½āˆ˜ī„š¶š‘–š‘—š›¾, (2.25) becomes š¶š›¼š›½š›¾=āˆ˜š¶š›¼š›½š›¾+īƒ©šœ‘ī…žī…žī‚€š›æšœ‘ā€²š‘–š‘—āˆ˜š‘¦š‘˜+š›æš‘–š‘˜āˆ˜š‘¦š‘—ī‚+šœ‘ī…žī…žšœ‘ā€²+2š¹2šœ‘ī…žī…žš‘”š‘—š‘˜š‘¦š‘–+2ī€·šœ‘ī…žī…žī…žšœ‘ā€²āˆ’2šœ‘ī…žī…ž2ī€øī€·šœ‘ā€²šœ‘ā€²+2š¹2šœ‘ī…žī…žī€øš‘¦š‘–āˆ˜š‘¦š‘—āˆ˜š‘¦š‘˜īƒŖšµš‘˜š›¾šµš›¼š‘–šµš‘—š›½.(2.26) Thus, we have the following.

Theorem 2.7. The coefficients of the induced tangent connection š·š‘‡ for the subspace ī‚šæš‘š of an APL space are given by (2.24) and (2.26).

Remarks 2. The torsion š‘‡š›¼š›½š›¾=šæš›¼š›½š›¾āˆ’šæš›¼š›¾š›½ does not vanish, in general, while š‘†š›¼š›½š›¾=š¶š›¼š›½š›¾āˆ’š¶š›¼š›¾š›½=0. These facts may be observed from (2.24) and (2.26).

Definition 2.8 (cf. [8]). An operator š·āŸ‚ given by š·āŸ‚š‘‹š‘Ž=š‘‹š‘Ž|š›¼š‘‘š‘¢š›¼+š‘‹š‘Ž|š›¼š›æš‘£š›¼,(2.27) where š‘‹š‘Ž|š›¼=š›æš›¼š‘‹š‘Ž+š‘‹š‘šæš‘Žš‘š›¼,š‘‹š‘Ž|š›¼=Ģ‡šœ•š›¼š‘‹š‘Ž+š‘‹š‘š¶š‘Žš‘š›¼, is called the induced normal connection by š·.
The coefficients (šæš‘Žš‘š›¾,š¶š‘Žš‘š›¾) of š·āŸ‚ are given by šæš‘Žš‘š›¾=šµš‘Žš‘–ī‚€š›æš›¾šµš‘–š‘+šµš‘—š‘ī‚šæš‘–š‘—š›¾ī‚,(2.28)š¶š‘Žš‘š›¾=šµš‘Žš‘–ī‚€Ģ‡šœ•š›¾šµš‘–š‘+šµš‘—š‘ī‚š¶š‘–š‘—š›¾ī‚.(2.29) Using (2.6) and (2.16) in (2.28), we find šæš‘Žš‘š›¾=šµš‘Žš‘–āˆ˜š›æš›¾šµš‘–š‘+šµš‘Žš‘–šµš›¼š‘š‘‰š‘š‘—šµš‘—š›¾Ģ‡šœ•š›¼šµš‘–š‘+šµš‘—š‘šµš‘Žš‘–īƒ¬āˆ˜ī„šæš‘–š‘—š›¾+ī‚µš‘‰š‘Ÿš‘˜š¶š‘–š‘—š‘Ÿ+š‘‰š‘Ÿš‘—š¶š‘–š‘˜š‘Ÿ+š‘‰š‘Ÿš‘š‘Žš‘–š‘š¶š‘Ÿš‘˜š‘—āˆ’āˆ˜š¶š‘–š‘—š‘Ÿšµš‘Ÿš‘šµš‘š‘š‘‰š‘š‘˜ī‚¶šµš‘˜š›¾+ī‚µšœ‘ī…žī…žšœ‘ā€²āˆ˜š‘¦š‘—š›æš‘–š‘˜+šœ‘ī…žī…žšœ‘ā€²+2š¹2šœ‘ī…žī…žš‘”š‘—š‘˜š‘¦š‘–ī‚¶šµš‘˜š‘š»š‘š›¾ī‚¹.(2.30) Taking āˆ˜šæš‘Žš‘š›¾=šµš‘Žš‘–(āˆ˜š›æš›¾šµš‘–š‘+šµš‘—š‘āˆ˜ī„šæš‘–š‘—š›¾) and using āˆ˜š‘¦š‘—šµš‘—š‘=0, (2.30) reduces to šæš‘Žš‘š›¾=āˆ˜šæš‘Žš‘š›¾+šµš‘Žš‘–šµš›¼š‘š‘‰š‘š‘—šµš‘—š›¾Ģ‡šœ•š›¼šµš‘–š‘+ī‚µš‘‰š‘Ÿš‘˜š¶š‘–š‘—š‘Ÿ+š‘‰š‘Ÿš‘—š¶š‘–š‘˜š‘Ÿ+š‘‰š‘Ÿš‘š‘Žš‘–š‘š¶š‘Ÿš‘˜š‘—āˆ’āˆ˜š¶š‘–š‘—š‘Ÿšµš‘Ÿš‘šµš‘š‘š‘‰š‘š‘˜ī‚¶šµš‘Žš‘–šµš‘—š‘šµš‘˜š›¾+šœ‘ī…žī…žšœ‘ā€²+2š¹2šœ‘ī…žī…žš‘”š‘—š‘˜š‘¦š‘–šµš‘˜š‘š»š‘š›¾šµš‘Žš‘–šµš‘—š‘.(2.31) Next, using (2.18) in (2.29), we have š¶š‘Žš‘š›¾=šµš‘Žš‘–īƒ©Ģ‡šœ•š›¾šµš‘–š‘+šµš‘—š‘āˆ˜ī„š¶š‘–š‘—š›¾īƒŖ+īƒ¬šœ‘ī…žī…žī‚€š›æšœ‘ā€²š‘–š‘—āˆ˜š‘¦š‘˜+š›æš‘–š‘˜āˆ˜š‘¦š‘—ī‚+šœ‘ī…žī…žšœ‘ā€²+2š¹2šœ‘ī…žī…žš‘”š‘—š‘˜š‘¦š‘–+2ī€·šœ‘ī…žī…žī…žšœ‘ā€²āˆ’2šœ‘ī…žī…ž2ī€øī€·šœ‘ā€²šœ‘ā€²+2š¹2šœ‘ī…žī…žī€øš‘¦š‘–āˆ˜š‘¦š‘—āˆ˜š‘¦š‘˜īƒ­šµš‘˜š›¾šµš‘Žš‘–šµš‘—š‘.(2.32) Taking āˆ˜š¶š‘Žš‘š›¾=šµš‘Žš‘–(Ģ‡šœ•š›¾šµš‘–š‘+šµš‘—š‘āˆ˜ī„š¶š‘–š‘—š›¾) and using (1.9) and āˆ˜š‘¦š‘—šµš‘—š‘=0, the last equation yields š¶š‘Žš‘š›¾=āˆ˜š¶š‘Žš‘š›¾+šœ‘ī…žī…žš›æšœ‘ā€²š‘Žš‘āˆ˜š‘¦š‘˜šµš‘˜š›¾+šœ‘ī…žī…žšœ‘ā€²+2š¹2šœ‘ī…žī…žš‘”š‘—š‘˜š‘¦š‘–šµš‘˜š›¾šµš‘Žš‘–šµš‘—š‘.(2.33) Thus, we have the following.

Theorem 2.9. The coefficients of induced normal connection š·āŸ‚ for the subspace ī‚šæš‘š of an APL space šæš‘› are given by (2.31) and (2.33).

Definition 2.10 (cf. [8]). The (mixed) derivative of a mixed d-tensor field š‘‡š‘–ā‹Æš›¼ā‹Æš‘Žš‘—ā‹Æš›½ā‹Æš‘ is given by āˆ‡š‘‡š‘–ā‹Æš›¼ā‹Æš‘Žš‘—ā‹Æš›½ā‹Æš‘=ī‚€š›æšœ‚š‘‡š‘–ā‹Æš›¼ā‹Æš‘Žš‘—ā‹Æš›½ā‹Æš‘+š‘‡š‘˜ā‹Æš›¼ā‹Æš‘Žš‘—ā‹Æš›½ā‹Æš‘īƒ‚šæš‘–š‘˜šœ‚+ā‹Æ+š‘‡š‘–ā‹Æš›¾ā‹Æš‘Žš‘—ā‹Æš›½ā‹Æš‘šæš›¼š›¾šœ‚+ā‹Æ+š‘‡š‘–ā‹Æš›¼ā‹Æš‘š‘—ā‹Æš›½ā‹Æš‘šæš‘Žš‘šœ‚āˆ’š‘‡š‘–ā‹Æš›¼ā‹Æš‘Žš‘˜ā‹Æš›½ā‹Æš‘ī‚šæš‘˜š‘—šœ‚āˆ’ā‹Æāˆ’š‘‡š‘–ā‹Æš›¼ā‹Æš‘Žš‘—ā‹Æš›¾ā‹Æš‘šæš›¾š›½šœ‚āˆ’ā‹Æāˆ’š‘‡š‘–ā‹Æš›¼ā‹Æš‘Žš‘—ā‹Æš›½ā‹Æš‘ī‚šæš‘š‘šœ‚ī‚š‘‘š‘¢šœ‚+ī‚€Ģ‡šœ•šœ‚š‘‡š‘–ā‹Æš›¼ā‹Æš‘Žš‘—ā‹Æš›½ā‹Æš‘+š‘‡š‘˜ā‹Æš›¼ā‹Æš‘Žš‘—ā‹Æš›½ā‹Æš‘ī‚š¶š‘–š‘˜šœ‚+ā‹Æ+š‘‡š‘–ā‹Æš›¾ā‹Æš‘Žš‘—ā‹Æš›½ā‹Æš‘š¶š›¼š›¾šœ‚+ā‹Æ+š‘‡š‘–ā‹Æš›¼ā‹Æš‘š‘—ā‹Æš›½ā‹Æš‘š¶š‘Žš‘šœ‚āˆ’š‘‡š‘–ā‹Æš›¼ā‹Æš‘Žš‘˜ā‹Æš›½ā‹Æš‘ī‚š¶š‘˜š‘—šœ‚āˆ’ā‹Æāˆ’š‘‡š‘–ā‹Æš›¼ā‹Æš‘Žš‘—ā‹Æš›¾ā‹Æš‘š¶š›¾š›½šœ‚āˆ’ā‹Æāˆ’š‘‡š‘–ā‹Æš›¼ā‹Æš‘Žš‘—ā‹Æš›½ā‹Æš‘ī‚š¶š‘š‘šœ‚ī‚š›æš‘£šœ‚.(2.34) The connection 1-forms, ī„šœ”š‘–š‘—ī‚šæ=āˆ¶š‘–š‘—š›¼š‘‘š‘¢š›¼+ī‚š¶š‘–š‘—š›¼š›æš‘£š›¼,(2.35)šœ”š›¼š›½=āˆ¶šæš›¼š›½š›¾š‘‘š‘¢š›¾+š¶š›¼š›½š›¾š›æš‘£š›¾,(2.36)šœ”š‘Žš‘=āˆ¶šæš‘Žš‘š›¾š‘‘š‘¢š›¾+š¶š‘Žš‘š›¾š›æš‘£š›¾,(2.37) are called the connection 1-forms of āˆ‡. We have the following structure equations of āˆ‡.

Theorem 2.11 (cf. [8]). The structure equations of āˆ‡ are as follows: š‘‘(š‘‘š‘¢š›¼)āˆ’š‘‘š‘¢š›½āˆ§šœ”š›¼š›½=āˆ’Ī©š›¼,š‘‘(š›æš‘¢š›¼)āˆ’š›æš‘¢š›½āˆ§šœ”š›¼š›½Ģ‡Ī©=āˆ’š›¼,š‘‘ī„šœ”š‘–š‘—āˆ’ī„šœ”ā„Žš‘—āˆ§ī„šœ”š‘–ā„Žī„Ī©=āˆ’š‘–š‘—,š‘‘šœ”š›¼š›½āˆ’šœ”š›¾š›½āˆ§šœ”š›¼š›¾=āˆ’Ī©š›¼š›½,š‘‘šœ”š‘Žš‘āˆ’šœ”š‘š‘āˆ§šœ”š‘Žš‘=āˆ’Ī©š‘Žš‘,(2.38) where the 2-forms of torsions Ī©š›¼,Ģ‡Ī©š›¼ are given by Ī©š›¼=12š‘‡š›¼š›½š›¾š‘‘š‘¢š›½āˆ§š‘‘š‘¢š›¾+š¶š›¼š›½š›¾š‘‘š‘¢š›½āˆ§š›æš‘£š›¾,Ģ‡Ī©š›¼=12š‘…š›¼š›½š›¾š‘‘š‘¢š›½āˆ§š‘‘š‘¢š›¾+š‘ƒš›¼š›½š›¾š‘‘š‘¢š›½āˆ§š›æš‘£š›¾,(2.39) with š‘ƒš›¼š›½š›¾=Ģ‡šœ•š›¾ī‚š‘š›¼š›½āˆ’šæš›¼š›½š›¾, and the 2-forms of curvature ī„Ī©š‘–š‘—,Ī©š›¼š›½ and Ī©š‘Žš‘, are given by ī„Ī©š‘–š‘—=12īƒ‚š‘…š‘–š‘—š›¼š›½š‘‘š‘¢š›¼āˆ§š‘‘š‘¢š›½+īƒ‚š‘ƒš‘–š‘—š›¼š›½š‘‘š‘¢š›¼āˆ§š›æš‘£š›½+12īƒ‚š‘†š‘–š‘—š›¼š›½š›æš‘£š›¼āˆ§š›æš‘£š›½,Ī©š›¼š›½=12š‘…š›¼š›½š›¾š›æš‘‘š‘¢š›¾āˆ§š‘‘š‘¢š›æ+š‘ƒš›¼š›½š›¾š›æš‘‘š‘¢š›¾āˆ§š›æš‘£š›æ+12š‘†š›¼š›½š›¾š›æš›æš‘£š›¾āˆ§š›æš‘£š›æ,Ī©š‘Žš‘=12š‘…š‘Žš‘š›¼š›½š‘‘š‘¢š›¼āˆ§š‘‘š‘¢š›½+š‘ƒš‘Žš‘š›¼š›½š‘‘š‘¢š›¼āˆ§š›æš‘£š›½+12š‘†š‘Žš‘š›¼š›½š›æš‘£š›¼āˆ§š›æš‘£š›½.(2.40)

We will use the following notations in Section 4: (a)ī„Ī©š‘–š‘—=ī„Ī©ā„Žš‘–š‘Žā„Žš‘—,(b)Ī©š›¼š›½=Ī©š›¾š›¼š‘Žš›¾š›½,(c)Ī©š‘Žš‘=Ī©š‘š‘š›æš‘Žš‘.(2.41)

3. The Gauss-Weingarten Formulae

The Gauss-Weingarten formulae for the subspace ī‚šæš‘šī‚ī„=(š‘€,šæ(š‘¢,š‘£)) of a Lagrange space šæš‘› are given by (cf. [8]) āˆ‡šµš‘–š›¼=šµš‘–š‘ŽĪ š‘Žš›¼,āˆ‡šµš‘–š‘Ž=āˆ’šµš‘–š›½Ī š›½š‘Ž,(3.1) where Ī š‘Žš›¼=š»š‘Žš›¼š›½š‘‘š‘¢š›½+š¾š‘Žš›¼š›½š›æš‘£š›½,Ī š›½š‘Ž=š‘”š›½š›¾š›æš‘Žš‘Ī š‘š›¾,(3.2)(a)š»š‘Žš›¼š›½=šµš‘Žš‘–ī‚€š›æš›½šµš‘–š›¼+šµš‘—š›¼ī‚šæš‘–š‘—š›½ī‚,(b)š¾š‘Žš›¼š›½=šµš‘Žš‘–šµš‘—š›¼ī‚š¶š‘–š‘—š›½.(3.3) Using (2.6) and (2.16) in (3.3)(a), we have š»š‘Žš›¼š›½=šµš‘Žš‘–īƒ©āˆ˜š›æš›½šµš‘–š›¼+šµš‘—š›¼āˆ˜ī„šæš‘–š‘—š›½īƒŖ+šµš‘Žš‘–šµš›¾š‘š‘‰š‘š‘—šµš‘—š›½šµš‘–š›¼š›¾+ī‚µš‘‰š‘Ÿš‘˜š¶š‘–š‘—š‘Ÿ+š‘‰š‘Ÿš‘—š¶š‘–š‘˜š‘Ÿ+š‘‰š‘Ÿš‘š‘Žš‘–š‘š¶š‘Ÿš‘˜š‘—āˆ’āˆ˜š¶š‘–š‘—š‘Ÿšµš‘Ÿš‘šµš‘š‘š‘‰š‘š‘˜ī‚¶šµš‘Žš‘–šµš‘—š›¼šµš‘˜š›½+ī‚µšœ‘ī…žī…žšœ‘ā€²āˆ˜š‘¦š‘—š›æš‘–š‘˜+šœ‘ī…žī…žšœ‘ā€²+2š¹2šœ‘ī…žī…žš‘”š‘—š‘˜š‘¦š‘–ī‚¶šµš‘˜š‘š»š‘š›½šµš‘Žš‘–šµš‘—š›¼.(3.4) If we take āˆ˜š»š‘Žš›¼š›½=šµš‘Žš‘–(āˆ˜š›æš›½šµš‘–š›¼+šµš‘—š›¼āˆ˜ī„šæš‘–š‘—š›½), the last expression provides š»š‘Žš›¼š›½=āˆ˜š»š‘Žš›¼š›½+šµš‘Žš‘–šµš›¾š‘š‘‰š‘š‘—šµš‘—š›½šµš‘–š›¼š›¾+ī‚µš‘‰š‘Ÿš‘˜š¶š‘–š‘—š‘Ÿ+š‘‰š‘Ÿš‘—š¶š‘–š‘˜š‘Ÿ+š‘‰š‘Ÿš‘š‘Žš‘–š‘š¶š‘Ÿš‘˜š‘—āˆ’āˆ˜š¶š‘–š‘—š‘Ÿšµš‘Ÿš‘šµš‘š‘š‘‰š‘š‘˜ī‚¶šµš‘Žš‘–šµš‘—š›¼šµš‘˜š›½+ī‚µšœ‘ī…žī…žšœ‘ā€²āˆ˜š‘¦š‘—š›æš‘–š‘˜+šœ‘ī…žī…žšœ‘ā€²+2š¹2šœ‘ī…žī…žš‘”š‘—š‘˜š‘¦š‘–ī‚¶šµš‘˜š‘š»š‘š›½šµš‘Žš‘–šµš‘—š›¼.(3.5) Next, using (2.18) in (3.3)(b) and keeping (1.9) in view, we find š¾š‘Žš›¼š›½=āˆ˜š¾š‘Žš›¼š›½+īƒ©šœ‘ī…žī…žšœ‘ā€²+2š¹2šœ‘ī…žī…žš‘”š‘—š‘˜š‘¦š‘–+2ī€·šœ‘ī…žī…žī…žšœ‘ā€²āˆ’šœ‘ī…žī…ž2ī€øī€·šœ‘ā€²šœ‘ā€²+2š¹2šœ‘ī…žī…žī€øš‘¦š‘–āˆ˜š‘¦š‘—āˆ˜š‘¦š‘˜īƒŖšµš‘Žš‘–šµš‘—š›¼šµš‘˜š›½,(3.6) where āˆ˜š¾š‘Žš›¼š›½=šµš‘Žš‘–šµš‘—š›¼āˆ˜ī„š¶š‘–š‘—š›½. Thus, we have the following.

Theorem 3.1. The following Gauss-Weingarten formulae for the subspace ī‚šæš‘š of an APL space hold: āˆ‡šµš‘–š›¼=šµš‘–š‘ŽĪ š‘Žš›¼,āˆ‡šµš‘–š‘Ž=āˆ’šµš‘–š›½Ī š›½š‘Ž,(3.7) where Ī š‘Žš›¼=š»š‘Žš›¼š›½š‘‘š‘¢š›½+š¾š‘Žš›¼š›½š›æš‘£š›½,Ī š›½š‘Ž=š‘”š›½š›¾š›æš‘Žš‘Ī š‘š›¾,š»š‘Žš›¼š›½=āˆ˜š»š‘Žš›¼š›½+šµš‘Žš‘–šµš›¾š‘š‘‰š‘š‘—šµš‘—š›½šµš‘–š›¼š›¾+ī‚µš‘‰š‘Ÿš‘˜š¶š‘–š‘—š‘Ÿ+š‘‰š‘Ÿš‘—š¶š‘–š‘˜š‘Ÿ+š‘‰š‘Ÿš‘š‘Žš‘–š‘š¶š‘Ÿš‘˜š‘—āˆ’āˆ˜š¶š‘–š‘—š‘Ÿšµš‘Ÿš‘šµš‘š‘š‘‰š‘š‘˜ī‚¶šµš‘Žš‘–šµš‘—š›¼šµš‘˜š›½+ī‚µšœ‘ī…žī…žšœ‘ā€²āˆ˜š‘¦š‘—š›æš‘–š‘˜+šœ‘ī…žī…žšœ‘ā€²+2š¹2šœ‘ī…žī…žš‘”š‘—š‘˜š‘¦š‘–ī‚¶šµš‘˜š‘š»š‘š›½šµš‘Žš‘–šµš‘—š›¼,š¾š‘Žš›¼š›½=āˆ˜š¾š‘Žš›¼š›½+īƒ©šœ‘ī…žī…žšœ‘ā€²+2š¹2šœ‘ī…žī…žš‘”š‘—š‘˜š‘¦š‘–+2ī€·šœ‘ī…žī…žī…žšœ‘ī…žāˆ’šœ‘ī…žī…ž2ī€øšœ‘ī…žī€·šœ‘ī…ž+2š¹2šœ‘ī…žī…žī€øš‘¦š‘–āˆ˜š‘¦š‘—āˆ˜š‘¦š‘˜īƒŖšµš‘Žš‘–šµš‘—š›¼šµš‘˜š›½.(3.8)

Remark 3.2. š»š‘Žš›¼š›½ and š¾š‘Žš›¼š›½ given, respectively, by (3.5) and (3.6) are called the second fundamental š‘‘-tensor fields of immersion š‘–.

The following consequences of Theorem 3.1 are straightforward.

Corollary 3.3. In a subspace ī‚šæš‘š of an APL space, we have the following: (a)āˆ‡š‘Žš›¼š›½(=0,b)āˆ‡šµš‘–š›¼=0,(3.9) if and only if āˆ˜š»š‘Žš›¼š›½ī‚øšµ=āˆ’š‘Žš‘–šµš›¾š‘š‘‰š‘š‘—šµš‘—š›½šµš‘–š›¼š›¾+ī‚µš‘‰š‘Ÿš‘˜š¶š‘–š‘—š‘Ÿ+š‘‰š‘Ÿš‘—š¶š‘–š‘˜š‘Ÿ+š‘‰š‘Ÿš‘š‘Žš‘–š‘š¶š‘Ÿš‘˜š‘—āˆ’āˆ˜š¶š‘–š‘—š‘Ÿšµš‘Ÿš‘šµš‘š‘š‘‰š‘š‘˜ī‚¶šµš‘Žš‘–šµš‘—š›¼šµš‘˜š›½+ī‚µšœ‘ī…žī…žšœ‘ī…žāˆ˜š‘¦š‘—š›æš‘–š‘˜+šœ‘ī…žī…žšœ‘ī…ž+2š¹2šœ‘ī…žī…žš‘”š‘—š‘˜š‘¦š‘–ī‚¶šµš‘˜š‘š»š‘š›½šµš‘Žš‘–šµš‘—š›¼ī‚¹,āˆ˜š¾š‘Žš›¼š›½āŽ›āŽœāŽœāŽšœ‘=āˆ’ī…žī…žšœ‘ā€²+2š¹2šœ‘ī…žī…žš‘”š‘—š‘˜š‘¦š‘–+2ī‚€šœ‘ī…žī…žī…žšœ‘ī…žāˆ’šœ‘2ī…žī…žī‚šœ‘ī…žī€·šœ‘ī…ž+2š¹2šœ‘ī…žī…žī€øš‘¦š‘–āˆ˜š‘¦š‘—āˆ˜š‘¦š‘˜āŽžāŽŸāŽŸāŽ šµš‘Žš‘–šµš‘—š›¼šµš‘˜š›½.(3.10)

4. The Gauss-Codazzi Equations

The Gauss-Codazzi Equations for the subspace ī‚šæš‘šī‚ī„=(š‘€,šæ(š‘¢,š‘£)) of a Lagrange space šæš‘› are given by (cf. [8]) šµš‘–š›¼šµš‘—š›½ī„Ī©š‘–š‘—āˆ’Ī©š›¼š›½=Ī š›½š‘Žāˆ§Ī š‘Žš›¼,(4.1)šµš‘–š‘Žšµš‘—š‘ī„Ī©š‘–š‘—āˆ’Ī©š‘Žš‘=Ī š›¾š‘āˆ§Ī š›¾š‘Ž,(4.2)āˆ’šµš‘–š›¼šµš‘—š‘Žī„Ī©š‘–š‘—=š›æš‘Žš‘ī‚€š‘‘Ī š‘š›¼+Ī š‘š›½āˆ§šœ”š›½š›¼āˆ’Ī š‘š›¼āˆ§šœ”š‘š‘ī‚,(4.3) where (a)Ī š›¼š‘Ž=š‘”š›¼š›½Ī š›½š‘Ž,(b)Ī š›¾š‘=š›æš‘š‘Ī š‘š›¾.(4.4) Using (1.3) in (2.41)(a), we find that ī„Ī©š‘–š‘—ī„Ī©=šœ‘ā€²ā„Žš‘–š‘”ā„Žš‘—+2šœ‘ī…žī…žī„Ī©ā„Žš‘–āˆ˜š‘¦ā„Žāˆ˜š‘¦š‘—.(4.5) Applying š‘Žš›¾š›½=šµš‘–š›¾šµš‘—š›½š‘Žš‘–š‘— in (2.41)(b), we have Ī©š›¼š›½=šµš‘–š›¾šµš‘—š›½Ī©š›¾š›¼š‘Žš‘–š‘—, which in view of (1.3) becomes Ī©š›¼š›½=šœ‘ā€²š‘”š‘–š‘—šµš‘–š›¾šµš‘—š›½Ī©š›¾š›¼+2šœ‘āˆ˜ī…žī…žš‘¦š‘–āˆ˜š‘¦š‘—šµš‘–š›¾šµš‘—š›½Ī©š›¾š›¼,(4.6) that is, Ī©š›¼š›½=šœ‘ā€²š‘”š›¾š›½Ī©š›¾š›¼+2šœ‘āˆ˜ī…žī…žš‘¦š‘–āˆ˜š‘¦š‘—šµš‘–š›¾šµš‘—š›½Ī©š›¾š›¼.(4.7) For the subspace ī‚šæš‘š of an APL space, (4.4)(a) is of the form Ī š›¼š‘Ž=š‘Žš›¼š›½Ī š›½š‘Ž, which in view of š‘Žš›¼š›½=šµš‘–š›¼šµš‘—š›½š‘Žš‘–š‘— and (1.3) becomes Ī š›¼š‘Ž=šœ‘ā€²šµš‘–š›¼šµš‘—š›½š‘Žš‘–š‘—Ī š›½š‘Ž+2šœ‘āˆ˜ī…žī…žš‘¦š‘–āˆ˜š‘¦š‘—šµš‘–š›¼šµš‘—š›½Ī š›½š‘Ž, that is, Ī š›¼š‘Ž=šœ‘ā€²š‘”š›¼š›½Ī š›½š‘Ž+2šœ‘āˆ˜ī…žī…žš‘¦š‘–āˆ˜š‘¦š‘—šµš‘–š›¼šµš‘—š›½Ī š›½š‘Ž.(4.8) Thus, we have the following.

Theorem 4.1. The Gauss-Codazzi equations for a Lagrange subspace ī‚šæš‘š of an APL space are given by (4.1)ā€“(4.3) with Ī š›¼š‘Ž, Ī š›¾š‘, ī„Ī©š‘–š‘—, Ī©š›¼š›½, and šœ”š‘š‘, respectively, given by (4.8), (4.4)(b), (4.5), (4.7), and (2.37).

Acknowledgments

Authors are thankful to the reviewers for their valuable comments and suggestions. S. K. Shukla gratefully acknowledges the financial support provided by the Council of Scientific and Industrial Research (CSIR), India.