§ 5 Preliminary Riemannian geometry
1. Riemann space
[ Riemann space and its metric tensor ] If there is a set of functions g ij ( x i )= g ji ( x i ) in the n -dimensional space R n , such that two adjacent points x i ,
The distance ds between x i + d x i is given by a positive definite quadratic form
d s 2 = g ij ( x )d x i d x j
Determined, then the space R n is called Riemann space, denoted as V n . The geometry in the Riemann space V n is called Riemann geometry . The quadratic type ds 2 is called the line element of V n . Differentiate into
And the arc length of any curve xi = xi ( t ) is the integral
because of the coordinate transformation
Next, ds 2 is an invariant, so
This shows that g ij ( x ) is a component of a second-order covariance tensor, which is called the metric tensor or fundamental tensor of the Riemann space V n .
[ 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
g ij a i = a j , g ij b i = b j
Then and are covariant vectors, their length and scalar product are respectively
g ij a i a j = a j a j , g ij a i b j = a j b j
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:
They are called Christopher's three-index symbols of the first and second types, respectively .
In addition, there is the equation
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 g ij , the Riemann connection is used to define the corresponding parallel movement called the Levy- Civita parallel movement .
Suppose a vector field a i = a i ( t ) is given along a certain curve x i = x i ( t ) in V n , if an infinitesimal displacement is made along this curve, the vector a i ( t ) follows the law
change, then the vector a i ( t ) is said to perform a Levi-Civita parallel translation along the curve .
The Levi-Civita parallel motion has the properties:
1 The covariant derivative of the metric tensor g ij is equal to zero, i.e.
2 If the two families of vectors a i ( t ) and b i ( t ) move parallel to the curve, then
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
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 R 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.
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 j , k and l , r; it is symmetric with respect to the former pair of indexes and the latter pair of indexes; after cyclic permutation of the first three indexes, the sum is equal to zero, that is,
R jklr = - R kjlr
R jklr = -R jkrl _
R jklr = R lrjk
R jklr + R kljr + R ljkr = 0
4 The Lich tensor is symmetric, that is, R kl = R lk .
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 , q i , also known as the section curvature .
If for all points in space V n ( n > 2) we have
R rijk = K ( g rk g ij − g rj g ik )
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 .