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.
- Riemannian product manifold
- Riemannian product structure
- integral manifold
- a distribution on a manifold
- real product space forms
- a slant distribution
Let be an immersion of an n-dimensional manifold into an m-dimensional Riemannian manifold . Denote by the induced Riemannian metric on . Thus, become an isometric immersion and is also a Riemannian manifold with the Riemannian metric for any vector fields in . The Riemannian metric on is called the induced metric on . In local components, with and .
If a vector field of at a point satisfies
for any vector of at , then is called a normal vector of in at . A unit normal vector field of in is called a normal section on .
By , we denote the vector bundle of all normal vectors of in . Then, the tangent bundle of is the direct sum of the tangent bundle of and the normal bundle of in ,
We note that if the sub-manifold is of codimension one in and they are both orientiable, we can always choose a normal section on ,
where is any arbitrary vector field on .
By denote the Riemannian connection on and we put
for any vector fields tangent to , where and are tangential and the normal components of , respectively. Formula is called the Gauss formula for the sub-manifold of a Riemannian manifold .
Proposition 1.1. ∇ is the Riemannian connection of the induced metric on and is a normal vector field over , which is symmetric and bilinear in and .
Proof: Let and be differentiable functions on . Then, we have
This implies that
Since the Riemannian connection has no torsion, we have
By comparing the tangential and normal parts of the last equality, we obtain
These equations show that has no torsion and is a symmetric bilinear map. Since the metric is parallel, we can easily see that
for any vector fields tangent to , that is, is also the Riemannian connection of the induced metric on .
We recall the second fundamental form of the sub-manifold (or immersion ), which is defined by
If identically, then sub-manifold is said to be totally geodesic, where is the set of the differentiable vector fields on normal bundle of .
Totally geodesic sub-manifolds are simplest sub-manifolds.
Definition 1.1. Let be an n-dimensional sub-manifold of an m-dimensional Riemannian manifold . By , we denote the second fundamental form of in .
is called the mean curvature vector of in . If , the sub-manifold is called minimal.
On the other hand, is called pseudo-umbilical if there exists a function on , such that
for any vector fields on and is called totally umbilical sub-manifold if
It is clear that every minimal sub-manifold is pseudo-umbilical with . On the other hand, by a direct calculation, we can find 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 .
Now, let be a sub-manifold of a Riemannian manifold and be a normal vector field on , be a vector field on . Then, we decompose
where and denote the tangential and the normal components of , respectively. We can easily see that and are both differentiable vector fields on and normal bundle of , respectively. Moreover, Eq. (15) is also called Weingarten formula.
Proposition 1.2. Let be a sub-manifold of a Riemannian manifold . Then
(a) is bilinear in vector fields and . Hence, at point depends only on vector fields and .
(b) For any normal vector field on , we have
Proof: Let and be any two functions on . Then, we have
This implies that
Thus, is bilinear in and . Additivity is trivial. On the other hand, since is a Riemannian metric,
for any and .
Eq. (12) implies that
The proof is completed .
Let be a sub-manifold of a Riemannian manifold , and and denote the second fundamental form and shape operator of respectively.
The covariant derivative of and is, respectively, defined by
for any vector fields tangent to and any vector field normal to . If for all , then the second fundamental form of is said to be parallel, which is equivalent to . By direct calculations, we get the relation
Example 1.1. We consider the isometric immersion
we note that is a two-dimensional sub-manifold of and the tangent bundle is spanned by the vectors
and the normal vector fields
By , we denote the Levi-Civita connection of , the coefficients of connection, are given by
Thus, we have , and The mean curvature vector of is given by
Furthermore, by using Eq. (16), we obtain
Thus, we have
Now, let be a sub-manifold of a Riemannian manifold , and be the Riemannian curvature tensors of and , respectively. From then the Gauss and Weingarten formulas, we have
for any vector fields and tangent to . For any vector field tangent to , Eq. (37) gives the Gauss equation
On the other hand, the normal component of Eq. (37) is called equation of Codazzi, which is given by
If the Codazzi equation vanishes identically, then sub-manifold is said to be curvature-invariant sub-manifold .
In particular, if is of constant curvature, is tangent to , 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 of the normal bundle of the sub-manifold by
for any vector fields tangent to sub-manifold , and any vector field normal to . From the Gauss and Weingarten formulas, we have
For any normal vector to , we obtain
Since , Eq. (42) implies
Eq. (43) is also called the Ricci equation.
If , then the normal connection of is said to be flat .
When , the normal connection of the sub-manifold is flat if and only if the second fundamental form is commutative, i.e. for all . If the ambient space is real space form, then and hence the normal connection of is flat if and only if the second fundamental form is commutative. If tangent to , then equation of codazzi Eq. (37) reduces to
which is equivalent to
On the other hand, if the ambient space is a space of constant curvature , then we have
for any vector fields and on .
Since is tangent to , the equation of Gauss and the equation of Ricci reduce to
Proposition 1.3. A totally umbilical sub-manifold in a real space form of constant curvature is also of constant curvature.
This shows that the sub-manifold is of constant curvature for . If , follows from the equation of Codazzi .
This proves the proposition.
On the other hand, for any orthonormal basis of normal space, we have
Thus, Eq. (45) can be rewritten as
Now, let - be the Ricci tensor of . Then, Eq. (47) gives us
where are orthonormal basis of .
Therefore, the scalar curvature of sub-manifold is given by
is the square of the length of the second fundamental form of , which is denoted by . Thus, we also have
2. Distribution on a manifold
An m-dimensional distribution on a manifold is a mapping defined on , which assignes to each point of an m-dimensional linear subspace of . A vector field on belongs to if we have for each . When this happens, we write . The distribution is said to be differentiable if for any , there exist m-differentiable linearly independent vector fields in a neighbordhood of p.
The distribution is said to be involutive if for all vector fields we have . A sub-manifold of is said to be an integral manifold of if for every point , coincides with the tangent space to at . If there exists no integral manifold of which contains , then is called a maximal integral manifold or a leaf of . The distribution is said to be integrable if for every , there exists an integral manifold of containing .
Let and distribution be a linear connection on , respectively. The distribution is said to be parallel with respect to , if we have
Now, let be Riemannian manifold and be a distribution on . We suppose is endowed with two complementary distribution and , we have . Denoted by and the projections of to and , respectively.
Theorem 2.1. All the linear connections with respect to which both distributions and are parallel, are given by
for any , where and are, respectively, an arbitrary linear connection and arbitrary tensor field of type on .
Proof: Suppose is an arbitrary linear connection on . Then, any linear connection on is given by
for any . We can put
for any . Then, we have
for any .
The distributions and are both parallel with respect to if and only if we have
Next, by means of the projections and , we define a tensor field of type on by
for any . By a direct calculation, it follows that . Thus, we say that defines an almost product structure on . The covariant derivative of is defined by
for all . We say that the almost product structure is parallel with respect to the connection , if we have . In this case, is called the Riemannian product structure .
Theorem 2.2. Let be a Riemannian manifold and , be orthogonal distributions on such that Both distributions and are parallel with respect to if and only if is a Riemannian product structure.
Proof: For any , we can write
that is, and so ,
In the same way, we obtain
which implies that
In the same way, we can find . Thus, we obtain
This proves our assertion .
Theorem 2.3. Both distributions and are parallel with respect to Levi-Civita connection if and only if they are integrable and their leaves are totally geodesic in .
Proof: Let us assume both distributions and are parallel. Since is a torsion free linear connection, we have
Thus, and are integrable distributions. Now, let be a leaf of and denote by the second fundamental form of the immersion of in . Then by the Gauss formula, we have
for any , where denote the Levi-Civita connection on . Since is parallel from Eq. (76) we conclude , that is, is totally in . In the same way, it follows that each leaf of is totally geodesic in .
Conversely, suppose and be integrable and their leaves are totally geodesic in . Then by using Eq. (4), we have
Since is a Riemannian metric tensor, we obtain
for any and Thus, both distributions and are parallel on .
3. Locally decomposable Riemannian manifolds
Let be dimensional Riemannian manifold and be a tensor type on such that , .
If the Riemannian metric tensor satisfying
for any then is called almost Riemannian product manifold and is said to be almost Riemannian product structure. If is parallel, that is, , then is said to be locally decomposable Riemannian manifold.
Now, let be an almost Riemannian product manifold. We put
Then, we have
Thus, and define two complementary distributions and globally. Since , we easily see that the eigenvalues of are and . An eigenvector corresponding to the eigenvalue is in and an eigenvector corresponding to is in . If has eigenvalue of multiplicity and eigenvalue of multiplicity , then the dimension of is and that of is . Conversely, if there exist in two globally complementary distributions and of dimension and , respectively. Then, we can define an almost Riemannian product structure on by by .
Let be a locally decomposable Riemannian manifold and we denote the integral manifolds of the distributions and by and , respectively. Then we can write , . Also, we denote the components of the Riemannian curvature of by , .
Now, we suppose that the two components are both of constant curvature and . Then, we have
Then, the above equations may also be written in the form
Conversely, suppose that the curvature tensor of a locally decomposable Riemannian manifold has the form
Then, we have
Let be an dimensional almost Riemannian product manifold with the Riemannian structure and be an dimensional sub-manifold of . For any vector field tangent to , we put
where and denote the tangential and normal components of , with respect to , respectively. In the same way, for , we also put
where and denote the tangential and normal components of , respectively.
Then, we have
On the other hand, we can easily see that
for any .
If for all , then is said to be invariant sub-manifold in , for each . In this case, and Thus, defines an almost product Riemannian on .
Conversely, is an almost product Riemannian structure on , the and hence is an invariant sub-manifold in .
Consequently, we can give the following theorem .
Theorem 3.1. Let be a sub-manifold of an almost Riemannian product manifold with almost Riemannian product structure . The induced structure on is an almost Riemannian product structure if and only if is an invariant sub-manifold of .
Definition 3.1. Let be a sub-manifold of an almost Riemannian product with almost product Riemannian structure . For each non-zero vector at , we denote the slant angle between and by . Then said to be slant sub-manifold if the angle is constant, it is independent of the choice of and .
Thus, invariant and anti-invariant immersions are slant immersions with slant angle and , respectively. A proper slant immersion is neither invariant nor anti-invariant.
Theorem 3.2. Let be a sub-manifold of an almost Riemannian product manifold with almost product Riemannian structure . is a slant sub-manifold if and only if there exists a constant , such tha
Furthermore, if the slant angle is , then it satisfies .
Definition 3.2. Let be a sub-manifold of an almost Riemannian product manifold with almost Riemannian product structure . is said to be semi-slant sub-manifold if there exist distributions and on such that
(i) has the orthogonal direct decomposition
(ii) The distribution is a slant distribution with slant angle
(iii) The distribution is an invariant distribution, .
In a semi-slant sub-manifold, if , then semi-slant sub-manifold is called semi-invariant sub-manifold .
Example 3.1. Now, let us consider an immersed sub-manifold in given by the equations
By direct calculations, it is easy to check that the tangent bundle of is spanned by the vectors
where and denote arbitrary parameters.
For the coordinate system of , we define the almost product Riemannian structure as follows:
Since and are orthogonal to and ,,are tangent to , we can choose a and . Thus, is a dimensional semi-invariant sub-manifold of with usual almost Riemannian product structure
Example 3.2. Let be sub-manifold of by given
where and are the arbitrary parameters. By direct calculations, we can easily see that the tangent bundle of is spanned by
For the almost Riemannian product structure of , is spanned by vectors
Since and are orthogonal to and and are tangent to , we can choose and . Thus, is a four-dimensional semi-invariant sub-manifold of with usual Riemannian product structure .
Definition 3.3. Let be a sub-manifold of an almost Riemannian product manifold with almost Riemannian product structure . is said to be pseudo-slant sub-manifold if there exist distributions and on such that
The tangent bundle .
The distribution is a slant distribution with slant angle .
The distribution is an anti-invariant distribution, .
As a special case, if and , then pseudo-slant sub-manifold becomes semi-invariant and anti-invariant sub-manifolds, respectively.
Example 3.3. Let be a sub-manifold of by the given equation
where and arbitrary parameters and is a constant.
We can check that the tangent bundle of is spanned by the tangent vectors
For the almost product Riemannian structure of whose coordinate systems choosing
Then, we have
Thus, is a slant distribution with slant angle . Since and are orthogonal to , is an anti-invariant distribution, that is, is a 4-dimensional proper pseudo-slant sub-manifold of with its almost Riemannian product structure