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The Lie Derivative of Normal Connections

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The Lie Derivative of Normal Connections Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 9 (2016), 4247{4256 Research Article The Lie derivative of normal connections Bui Cao Van Department of Mathematics, Vinh University, 182 Le Duan, Vinh City, Vietnam. Communicated by Y.-Z. Chen Abstract In this paper, we state the Lie derivative of normal connection on a submanifold M of the Riemannian manifold Mf. By this vein, we introduce the Lie derivative of the normal curvature tensor on M and give some relations between the normal curvature tensor on M and curvature tensor on Mf in the sense of the Lie derivative of normal connection. As an application, we give some detailed description of the normal curvature tensor on M whether M is a hypersubface. c 2016 All rights reserved. Keywords: Lie derivative, normal connection, normal curvature. 2010 MSC: 53C40, 53C15, 53C25. 1. Introduction The Lie derivative on differential forms is important operation. This is a generalization of the notion of directional derivative of a function. The Lie differentiation theory plays an important role in studying automorphisms of differential geometric structures. Moreover, the Lie derivative also is an essential tool in the Riemannian geometry. The Lie derivative of forms and its application was investigated by many authors (see [2{4, 6{8, 10] and the references given therein). In 2010, Sultanov used the Lie derivative of the linear connection to study the curvature tensor and the sorsion tensor on linear algebras [8]. In 2012, by invoking the Lie derivative of forms on the Riemannian n−dimensional manifold, the authors of [1] constructed the Lie derivative of the currents on Riemannian manifolds and given some applications on Lie groups. Recently, B. C. Van and T. T. K. Ha studied some properties of the Lie derivative of the linear connection r, the conjugate derivative dr with the linear n connection and using them for searching the curvature, the torsion of a space R along the linear flat connection r [9]. The main goal of the present work is to investigate some properties on the Lie derivative of the normal connection r? and of the normal curvature tensor R? on the submanifold M. We give some relations Email address: [email protected] (Bui Cao Van) Received 2016-04-16 B. C. Van, J. Nonlinear Sci. Appl. 9 (2016), 4247{4256 4248 between the normal curvature tensor on M and curvature tensor on Mf in the sense of the Lie derivative of normal connection. As an application, we give some detailed description of the normal curvature tensor on M whether M is a hypersubface. 2. Preliminaries Let M be an n-dimensional submanifold of an m−dimensional Riemannian manifold Mf equipped with a Riemannian metric ge. We denote the vector space of all smooth vector fields on M and Mf by B(M) and B(Mf), respectively. We denote 5e , 5, and 5? are the Levi-Civita, induced Levi-Civita induced normal connections in Mf, M, and the normal bundle N(M) of M, respectively. We use the inner product notation h; i ( or ·) for both the metrics ge of Mf and the induced metric g on the submanifold M. L At each p 2 M, the ambient tangent space TpMf splits as an orthogonal direct sum TpMf = TpM NpM, ? where NpM := (TpM) is the normal space at p with respect to the inner product g on TpMf. The set S e N(M) = NpM is called the normal bundle of M. If X; Y are vector fields in B(M), we can extend them p2M to vector fields on Mf, apply the ambient covariant derivative operator 5e and then decompose at points of M to get > ? 5e X Y = (5e X Y ) + (5e X Y ) : (2.1) The Gauss and Weingarten formulas are given respectively by (see [5], pp. 135) ? 5e X Y = 5X Y + σ(X; Y ) and 5e X N = −AN X + 5X N (2.2) for all X; Y 2 B(M) and N 2 N(M), where σ is the second fundamental form of M from B(M) × B(M) to N(M) given by ? σ(X; Y ) := (5e X Y ) ; (2.3) where X and Y are extended arbitrarily to Mf and the shape operator AN : X 7! AN X for all X 2 B(M);N 2 N(M). The Weingarten equation is given by (see [5], pp. 136) D E 5e X N; Y = − hN; σ(X; Y )i : (2.4) Thus, σ is the second fundamental form related to the shape operator A by hσ(X; Y );Ni = hAN X; Y i : (2.5) The equation of Gauss is given by (see [5], pp. 136) Re(X; Y; Z; W ) = R(X; Y; Z; W ) − hσ(X; W ); σ(Y; Z)i + hσ(X; Z); σ(Y; W )i (2.6) for all X; Y; Z; W 2 B(M), where Re and R are the Riemann curvature tensors of Mf and M, respectively. The curvature tensor R? of the normal bundle of M is defied by ? ? ? ? ? ? R (X; Y )N = 5X 5Y N − 5Y 5X N − 5[X;Y ]N (2.7) for any X; Y 2 B(M) and N 2 N(M). If R? = 0, then the normal connection 5? of M is said to be flat. 1 The mean curvature vector H is given by H = trace(σ): The submanifold M is totally geodesic in Mf n if σ = 0, and minimal if H = 0. Let N 2 N(M); the Weingarten map hN : B(M) ! B(M) is given by > hN (X) = − 5e X N (2.8) B. C. Van, J. Nonlinear Sci. Appl. 9 (2016), 4247{4256 4249 for all X 2 B(M): We easily get the following properties of the Weingarten mapping hN hN (X + Y ) = hN (X) + hN (Y ) ; hN ('X) = 'hN (X) (2.9) and hN ('X) :Y = 'hN (Y ) :X (2.10) for all X; Y 2 B(M);' 2 F(M): Next, we define the derivative of the Weingarten mapping hN . The map 5e hN : B(M) ! B(M) X (2.11) Y 7! (5e X hN )(Y ) = 5e X (hN (Y )) − hN (5e X Y ) is called the derivative of hN along a vector field X: ? Now, we define the Weingarten normal mapping and the derivative of the map hN along a vector field X. h? : B(M) ! N(M) N (2.12) ? ? X 7! hN (X) = 5X N is called the Weingarten normal mapping. We get the following properties (2.13), (2.14) of Weingarten ? normal mapping hN ? ? ? hN (X + Y ) = hN (X) + hN (Y ) 8X; Y 2 B(M): (2.13) ? ? hN ('X) = 'hN (X) 8X 2 B(M); 8' 2 F(M): (2.14) Let N; K 2 N(M) and ' 2 F(M); we obtain ? ? ? ? ? hN+K = hN + hK ; and h'N = 'hN : (2.15) ? Next, the derivative of the mapping hN along a vector field X is the mapping ? 5X h : B(M) ! N(M) N (2.16) ? ? ? ? Y 7! (5X hN )(Y ) = 5X (hN (Y )) − hN (5X Y ): ? ? We easily get the mapping hN and 5X hN are modular homomorphics. 3. Main results We begin at introducing the concept of Lie derivative of the normal vector field along a vector field. Definition 3.1. Suppose that X 2 B(M): The mapping ? LX : N(M) ! N(M) ? (3.1) ? ? N 7! LX N = [X; N] = 5e X N − 5e N X is called the Lie derivative of the normal vector field N along a vector field X. Definition 3.2. Given X 2 B(M) and r? is the normal connection on M. The mapping ? ? LX r : B(M) × N(M) ! N(M) ? ? ? ? ? ? ? (Y; N) 7! (LX r )(Y; N) = LX (rY N) − r[X;Y ]N − rY (LX N) is called the Lie derivative of the normal connection r? along a vector field X. B. C. Van, J. Nonlinear Sci. Appl. 9 (2016), 4247{4256 4250 For any X; Y 2 B(M); we have ? ? ? ? ? L[X;Y ] = LX oLY − LY oLX : (3.2) Let X; Y 2 B(M) and N 2 N(M): By using the definition on the Lie derivative of the normal connection, we obtain h ? ?i ? ? ? ? LX ; rY N = LX (rY N) − rY (LX N) ? ? ? = r[X;Y ]N + LX r (Y; N): Using this fact, we have the following formula, Proposition 3.3. If X; Y 2 B(M), then ? ? ? ? ? ? ? ? L[X;Y ]r = LX (LY r ) − LY (LX r ): Proof. Suppose that Z 2 B(M) and N 2 N(M); we have ? ? ? ? ? ? ? ? ? ? LX (LY r ) (Z; N) = LX ( LY r (Z; N)) − LY r ([X; Z] ;N) − LY r (Z; LX N) ? ? ? ? ? ? ? ? = LX (LY (rZ N)) − LX (rZ (LY N)) − LX (r[Y;Z]N) ? ? ? ? ? − LY (r[X;Z]N) + r[X;Z](LY N) + r[Y;[X;Z]]N ? ? ? ? ? ? ? ? − LY (rZ (LX N)) + rZ (LY (LX N)) + r[Y;Z](LX N) ? ? ? ? ? ? ? = LX (LY (rZ N)) − LX (r[Y;Z]N) − LY (r[X;Z]N) ? ? ? ? ? ? ? ? − LX (rZ (LY N)) − LY (rZ (LX N)) + r[X;Z](LY N) ? ? ? ? ? ? + r[Y;Z](LX N) + rZ (LY (LX N)) + r[Y;[X;Z]]N: Similarity, we have ? ? ? ? ? ? ? ? ? ? LY (LX r ) (Z; N) = LY (LX (rZ N)) − LY (r[X;Z]N) − LX (r[Y;Z]N) ? ? ? ? ? ? ? ? − LY (rZ (LX N)) − LX (rZ (LY N)) + r[Y;Z](LX N) ? ? ? ? ? ? + r[X;Z](LY N) + rZ (LX (LY N)) + r[X;[Y;Z]]N: Therefore, ? ? ? ? ? ? ? ? ? ? ? (LX (LY r ))(Z; N) − (LY (LX r ))(Z; N) = L[X;Y ](rZ N) − rZ (L[X;Y ]N) − r[[X;Y ];Z]N ? ? = L[X;Y ]r (Z; N); 8Z 2 B(M);N 2 N(M): ? ? ? ? ? ? ? ? So that L[X;Y ]r = LX (LY r ) − LY (LX r ): This proves the proposition.
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