Sub-Manifolds of a Riemannian Manifold Sub-Manifolds of a Riemannian Manifold

In this chapter, we introduce the theory of sub-manifolds of a Riemannian manifold. The fundamental notations are given. The theory of sub-manifolds of an almost Riemannian product manifold is one of the most interesting topics in differential geometry. According to the behaviour of the tangent bundle of a sub-manifold, with respect to the action of almost Riemannian product structure of the ambient manifolds, we have three typical classes of sub-manifolds such as invariant sub-manifolds, anti-invariant sub-manifolds and semi-invariant sub-manifolds. In addition, slant, semi-slant and pseudo-slant sub-manifolds are introduced by many geometers.


Introduction
Let i : M !M be an immersion of an n-dimensional manifold M into an m-dimensional Riemannian manifold ðM,gÞ. Denote by g ¼ i Ãg the induced Riemannian metric on M. Thus, i become an isometric immersion and M is also a Riemannian manifold with the Riemannian metric gðX, YÞ ¼gðX, YÞ for any vector fields X, Y in M. The Riemannian metric g on M is called the induced metric on M. In local components, g ij ¼ g AB B B j B A i with g ¼ g ji dx j dx j and g ¼ g BA dU B dU A .
If a vector field ξ p ofM at a point p∈M satisfies We note that if the sub-manifold M is of codimension one inM and they are both orientiable, we can always choose a normal section ξ on M, i:e:, gðX, ξÞ ¼ 0, gðξ, ξÞ ¼ 1, where X is any arbitrary vector field on M.
By∇, denote the Riemannian connection onM and we put for any vector fields X, Y tangent to M, where ∇ X Y and hðX, YÞ are tangential and the normal components of∇ X Y, respectively. Formula ð4Þ is called the Gauss formula for the sub-manifold M of a Riemannian manifold ðM,gÞ.
Proposition 1.1. ∇ is the Riemannian connection of the induced metric g ¼ i Ãg on M and hðX, YÞ is a normal vector field over M, which is symmetric and bilinear in X and Y.
Proof: Let α and β be differentiable functions on M. Then, we havẽ This implies that and hðαX, βYÞ ¼ αβhðX, YÞ: Eq. (6) shows that ∇ defines an affine connection on M and Eq. (4) shows that h is bilinear in X and Y since additivity is trivial [1].
Since the Riemannian connection∇ has no torsion, we have 0 ¼∇ X Y−∇ Y X−½X, Y ¼ ∇ X Y þ hðX, YÞ−∇ X Y−hðY, XÞ−½X, Y: By comparing the tangential and normal parts of the last equality, we obtain and hðX, YÞ ¼ hðY, XÞ: These equations show that ∇ has no torsion and h is a symmetric bilinear map. Since the metric g is parallel, we can easily see that for any vector fields X, Y, Z tangent to M, that is, ∇ is also the Riemannian connection of the induced metric g on M.
We recall h the second fundamental form of the sub-manifold M (or immersion i), which is defined by h : ΓðTMÞ · ΓðTMÞ ! ΓðT ⊥ MÞ: If h ¼ 0 identically, then sub-manifold M is said to be totally geodesic, where ΓðT ⊥ MÞ is the set of the differentiable vector fields on normal bundle of M.
Totally geodesic sub-manifolds are simplest sub-manifolds.
It is clear that every minimal sub-manifold is pseudo-umbilical with λ ¼ 0. On the other hand, by a direct calculation, we can find λ ¼gðH, HÞ for a pseudo-umbilical sub-manifold. So, every totally umbilical sub-manifold is a pseudo-umbilical and a totally umbilical sub-manifold is totally geodesic if and only if it is minimal [2]. Now, let M be a sub-manifold of a Riemannian manifold ðM,gÞ and V be a normal vector field on M, X be a vector field on M. Then, we decomposẽ where A V X and ∇ ⊥ X V denote the tangential and the normal components of ∇ ⊥ X V, respectively. We can easily see that A V X and ∇ ⊥ X V are both differentiable vector fields on M and normal bundle of M, respectively. Moreover, Eq. ð15Þ is also called Weingarten formula.
Proof: Let α and β be any two functions on M. Then, we havẽ This implies that Thus, A V X is bilinear in V and X. Additivity is trivial. On the other hand, since gis a Riemannian metric, for any X, Y∈ΓðTMÞ and V∈ΓðT ⊥ MÞ.
Let M be a sub-manifold of a Riemannian manifold ðM,gÞ, and h and A V denote the second fundamental form and shape operator of M, respectively.
The covariant derivative of h and A V is, respectively, defined by ð∇ X hÞðY, ZÞ ¼ ∇ ⊥ X hðY, ZÞ−hð∇ X Y, ZÞ−hðY, ∇ X ZÞ (23) and for any vector fields X, Y tangent to M and any vector field V normal to M. If ∇ X h ¼ 0 for all X, then the second fundamental form of M is said to be parallel, which is equivalent to ∇ X A ¼ 0.
By direct calculations, we get the relation Example 1.1. We consider the isometric immersion ffiffiffiffiffiffiffiffiffi ffi we note that M ¼ φðR 2 Þ⊂R 4 is a two-dimensional sub-manifold of R 4 and the tangent bundle is spanned by the vectors o and the normal vector fields By∇, we denote the Levi-Civita connection of R 4 , the coefficients of connection, are given bỹ and ∇ e2 e 1 ¼ 0: Furthermore, by using Eq. (16), we obtain and Thus, we have Now, let M be a sub-manifold of a Riemannian manifold ðM, gÞ,R and R be the Riemannian curvature tensors ofM and M, respectively. From then the Gauss and Weingarten formulas, we haveR If the Codazzi equation vanishes identically, then sub-manifold M is said to be curvatureinvariant sub-manifold [4].
In particular, ifM is of constant curvature,RðX, YÞZ is tangent to M, that is, sub-manifold is curvature-invariant. Whereas, in Kenmotsu space forms, and Sasakian space forms, this not true.
Next, we will define the curvature tensor R ⊥ of the normal bundle of the sub-manifold M by for any vector fields X, Y tangent to sub-manifold M, and any vector field V normal to M. From the Gauss and Weingarten formulas, we havẽ For any normal vector U to M, we obtain Eq. (43) is also called the Ricci equation.
If R ⊥ ¼ 0, then the normal connection of M is said to be flat [2].
which is equivalent to On the other hand, if the ambient spaceM is a space of constant curvature c, then we havẽ for any vector fields X, Y and Z onM. and respectively.
where fe 1 , e 2 , …, e n g are orthonormal basis of M.
Therefore, the scalar curvature r of sub-manifold M is given by

Distribution on a manifold
for any X, Y∈ΓðTMÞ, where ∇ 0 and S are, respectively, an arbitrary linear connection and arbitrary tensor field of type ð1, 2Þ onM.
Proof: Suppose ∇ 0 is an arbitrary linear connection onM. Then, any linear connection ∇ onM is given by for any X, Y∈ΓðTMÞ. We can put for any X∈ΓðTMÞ. Then, we have for any X, Y∈ΓðTMÞ. Next, by means of the projections P and Q, we define a tensor field F of type ð1, 1Þ onM by for any X∈ΓðTMÞ. By a direct calculation, it follows that F 2 ¼ I. Thus, we say that F defines an almost product structure onM. The covariant derivative of F is defined by for all X, Y∈ΓðTMÞ. We say that the almost product structure F is parallel with respect to the connection ∇, if we have ∇ X F ¼ 0. In this case, F is called the Riemannian product structure [2]. In the same way, we can find∇Q ¼ 0. Thus, we obtaiñ ∇F ¼∇ðP−QÞ ¼ 0: This proves our assertion [2].
for any X, Y∈ΓðDÞ, where ∇ 0 denote the Levi-Civita connection on M. Since D is parallel from Eq. (76) we conclude h ¼ 0, that is, M is totally inM. In the same way, it follows that each leaf of D ⊥ is totally geodesic inM.
Conversely, suppose D and D ⊥ be integrable and their leaves are totally geodesic inM. Then by using Eq. (4), we have ∇ X Y∈ΓðDÞ for any X, Y∈ΓðDÞ and ∇ U V∈ΓðD ⊥ Þ for any U, V∈ΓðD ⊥ Þ: Since g is a Riemannian metric tensor, we obtain and for any X, Y∈ΓðDÞ and U, V∈ΓðD ⊥ Þ: Thus, both distributions D and D ⊥ are parallel onM.

Locally decomposable Riemannian manifolds
Let ðM,gÞ be n−dimensional Riemannian manifold and F be a tensor ð1, 1Þ−type onM such that F 2 ¼ I, F≠∓I.
If the Riemannian metric tensorg satisfying gðX, YÞ ¼gðFX, FYÞ for any X, Y∈ΓðTMÞ thenM is called almost Riemannian product manifold and F is said to be almost Riemannian product structure. If F is parallel, that is, ð∇ X FÞY ¼ 0, thenM is said to be locally decomposable Riemannian manifold. Now, letM be an almost Riemannian product manifold. We put Then, we have Thus, P and Q define two complementary distributions P and Q globally. Since F 2 ¼ I, we easily see that the eigenvalues of F are 1 and −1. An eigenvector corresponding to the eigenvalue 1 is in P and an eigenvector corresponding to −1 is in Q. If F has eigenvalue 1 of multiplicity P and eigenvalue −1 of multiplicity q, then the dimension of P is p and that of Q is q. Conversely, if there exist inM two globally complementary distributions P and Q of dimension p and q, respectively. Then, we can define an almost Riemannian product structure F onM byM by F ¼ P−Q [7].
Let ðM,g, FÞ be a locally decomposable Riemannian manifold and we denote the integral manifolds of the distributions P and Q by M p and M q , respectively. Then we can writẽ M ¼ M p XM q , ðp, q > 2Þ. Also, we denote the components of the Riemannian curvature R ofM by R dcba , 1≤a, b, c, d≤n ¼ p þ q.
Now, we suppose that the two components are both of constant curvature λ and μ. Then, we have and R zyxw ¼ μfg zw g yx −g yw g zx g: Then, the above equations may also be written in the form ðλ−μÞfðF kh g ji −F jh g ki Þ þ ðg kh F ji −g jh F ki Þg: Conversely, suppose that the curvature tensor of a locally decomposable Riemannian manifold has the form R kjih ¼ afðg kh g ji −g jh g ki Þ þ ðF kh F ji −F jh F ki Þg þbfðF kh g ji −F jh g ki Þ þ ðg kh F ji −g jh F ki Þg: Then, we have and R zyxw ¼ 2ða−bÞfg zw g yx −g yw g zx g: LetM be an m−dimensional almost Riemannian product manifold with the Riemannian structure ðF,gÞ and M be an n−dimensional sub-manifold ofM. For any vector field X tangent to M, we put where f X and wX denote the tangential and normal components of FX, with respect to M, respectively. In the same way, for V∈ΓðT ⊥ MÞ, we also put where BV and CV denote the tangential and normal components of FV, respectively.
Then, we have On the other hand, we can easily see that gðX, f YÞ ¼ gð f X, YÞ (94) and gðX, YÞ ¼ gð f X, f YÞ þ gðwX, wYÞ for any X, Y∈ΓðTMÞ [6]. Conversely, ðf , gÞ is an almost product Riemannian structure on M, the w ¼ 0 and hence M is an invariant sub-manifold inM.
Example 3.1. Now, let us consider an immersed sub-manifold M in R 7 given by the equations By direct calculations, it is easy to check that the tangent bundle of M is spanned by the vectors where θ, β and u denote arbitrary parameters.
For the coordinate system of R 7 ¼ fðx 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 Þjx i ∈R, 1 ≤ i ≤ 7g, we define the almost product Riemannian structure F as follows: Since Fz 1 and Fz 3 are orthogonal to M and Fz 2 ,Fz 4 ,Fz 5 are tangent to M, we can choose a D ¼ S p fz 2 , z 4 , z 5 g and D ⊥ ¼ S p fz 1 , z 3 g. Thus, M is a 5−dimensional semi-invariant sub-manifold of R 7 with usual almost Riemannian product structure ðF, < , >Þ: Example 3.2. Let M be sub-manifold of R 8 by given where u, v and β are the arbitrary parameters. By direct calculations, we can easily see that the tangent bundle of M is spanned by For the almost Riemannian product structure F of R 8 ¼ R 4 xR 4 , FðTMÞ is spanned by vectors ii. The distribution D θ is a slant distribution with slant angle θ.
As a special case, if θ ¼ 0 and θ ¼ π 2 , then pseudo-slant sub-manifold becomes semi-invariant and anti-invariant sub-manifolds, respectively. where u, v, s and t arbitrary parameters and θ is a constant.
We can check that the tangent bundle of M is spanned by the tangent vectors For the almost product Riemannian structure F of R 6 whose coordinate systems ðx 1 , y 1 , x 2 , y 2 , x 3 , y 3 Þ choosing Then, we have Thus, D θ ¼ S p fe 1 , e 2 g is a slant distribution with slant angle α ¼ π 4 . Since Fe 3 and Fe 4 are orthogonal to M, D ⊥ ¼ S p fe 3 , e 4 g is an anti-invariant distribution, that is, M is a 4-dimensional proper pseudo-slant sub-manifold of R 6 with its almost Riemannian product structure ðF, < , >Þ: