**§ ****5 ****Preliminary Riemannian geometry**

1.
Riemann space

[ Riemann space and its metric tensor ] If there is a set of functions *g ** _{ij}* (

* ^{}*The distance

d *s *^{2
} = *g _{ij}*

Determined, then the space *R ^{n}* is called Riemann space, denoted as

_{}

And the arc length of any curve *xi ^{}* =

_{}

because of the coordinate transformation

_{}

Next, *ds *^{2} is an invariant, so

_{}

This shows that *g _{ij}* (

[ Length of vector, scalar product and angle of two vectors, adjoint tensor ] The definitions of scalar ( field ) , vector ( field ) , tensor ( field ) , etc. in Riemann space are similar to the previous sections, and their operations The rules are also similar .^{}

Let it be a contravariant vector, then the square of its length is_{}

*g _{ij} a ^{i} a ^{j}*

If and are two contravariant vectors, their scalar product is_{}_{}

*g _{ij} a ^{i} b ^{j}*

The cosine of the angle between the two vectors is

_{}

Assume

*g _{ij} a ^{i}* =

Then and are covariant vectors, their length and scalar product are respectively_{}_{}

*g _{ij} a ^{i}*

The adjoint tensor of the tensor is_{}

_{},_{}

where *g ^{lj}* satisfies the equation

^{}

where is the Kronecker notation ._{}^{}

[ Riemann contact and Christopher symbol ] In Riemann space, the contact can always be determined in a unique way , satisfying the conditions:_{}

(i) Affine connections are torsion-free, i.e._{}

(ii) The parallel movement generated by the affine connection keeps the length of the vector constant .

This is called the Riemann connection or the Levy-Civita connection ._{}

According to the above two conditions, it can be obtained that

_{}_{}

If you remember

_{}

then there are

_{}

Sometimes the following notation is used:

_{}and_{}

They are called Christopher's three-index symbols of the first and second types, respectively .

In addition, there is the equation

_{}

or

_{}

It should also be pointed out that all the results in § 4 concerning the covariant differential method hold for the Riemann connection ._{}

2.
The parallelism of Levi-Civita

The parallel movement in the affine connection space is determined by the affine connection . In the Riemann space *V ** ^{n}* with the metric tensor

Suppose a vector field *a ** ^{i}* =

_{}

change, then the vector *a ^{i}* (

The Levi-Civita parallel motion has the properties:

1 The covariant derivative of the metric tensor *g *_{ij is equal to zero, i.e.}^{ }_{}

_{}

Also ,
_{} _{}

2 ^{ }If the two families of vectors *a ^{i}* (

_{}

Therefore, the scalar product of the two vectors and the included angle remain unchanged under parallel translation .

3 ^{ }The self-parallel curves ( also called geodesics ) in the Riemann space *V ^{n}* are exactly the same as the self-parallel curves in the affine connection space, both of which are determined by the differential equation

_{}

But here is the Riemann connection . So a necessary and sufficient condition for a curve to be a geodesic is that its unit tangent vectors are parallel to each other ._{}_{}

3.
Curvature in Riemann space

[ Curvature tensor and Lich's formula ] The obvious difference between the covariant derivative of a tensor and the ordinary derivative is that when a higher-order derivative is obtained, the result of the tensor derivative is generally related to the order of the derivative . For example, when an operation acts on a vector , then there are_{}_{}

_{} (1)

remember

_{}

It is a third-order covariant and first-order contravariant fourth-order mixed tensor, called the curvature tensor or Riemann-Christopher tensor of space *V ^{n}* .

_{}

The left side is called the staggered second-order covariant derivative of the contravariant vector ; the staggered second-order covariant derivative with respect to the covariant vector is_{}_{}

_{}

The interleaved second-order covariant derivative of a tensor is

_{}

This is called the Leach formula .

[ Riemann notation · Lich tensor · Curvature scalar · Einstein space ]

covariate component of curvature tensor

_{}

are called Riemann symbols of the first kind, and Riemann symbols of the second kind ._{}

The curvature tensor shrinks to get

_{}

is called the Ricci tensor . The Ricci tensor is then contracted to get

*R* = *g ^{kl}*

called the curvature scalar .

If the Leach tensor satisfies

_{}

This space is called Einstein space .

[ Properties of Curvature Tensors ]

1 The first two indices *j* and *k* of the curvature tensor are antisymmetric, i.e.^{ }

_{}

very

_{}

2 ^{} The curvature tensor performs cyclic permutation of the three covariant indicators and adds them, so that

_{}

This is called the Lich identity .

3 ^{} The first kind of Riemann symbol *R _{kjlr}* can be calculated as follows:

_{}

Therefore , *R _{kjlr}* is antisymmetric with respect to indexes

*R _{jklr}* = -

*R _{jklr}* =

*R _{jklr}* =

*R _{jklr}* +

4 The Lich tensor is symmetric, that is, *R ** _{kl}* =

At any point in the 5 ^{} space *V ^{n}* , the following formula holds:

^{}

This is called the Pianchi identity . It states that the sum obtained by cyclic permutation of the index ( *i* ) of the covariant derivative and the first two indices ( *j* , *k* ) of the curvature tensor is equal to zero .

[ Riemann curvature ( section curvature ) and constant curvature space ] do the sum of two linearly independent vectors at a point *M* in the Riemann space *V ^{n}*

_{}

This is called the Riemannian curvature of the plane determined by *p ^{i}* ,

If for all points in space *V ^{n}* (

*R _{rijk}* =

Then the Riemann curvature *K* is a constant, which is Schur 's theorem .

*A space V ** ^{n}* with a constant Riemann curvature is called a space of constant curvature, and the line elements of this space can be transformed into the form

_{}

This is called a measure of a space of constant curvature in Riemann form .

A space with constant curvature is an Einstein space .

Contribute a better translation