Riemann geometry deals with geometric objects in non-linear space, such as the surface of the sphere etc.
First of all let's define some objects in such a space. Let's consider a transformation of coordinates:

*y ^{i}=y^{i}(x^{1},...,x^{n})*

If the coordinates of some vector

that is

this vector

that is

this vector

Tensor is formally written as the matrix, which components *C _{ij}* is the multiplied components
of some two vectors:

So contravariant tensor

In Riemann geometry Einstein's agreement of twin indexes sum is restricted:
the sum can only be obtained from covariant and contravariant indexes. For example:

*( A*B)=A^{i}*B_{i}*

is the scalar multiplication of vectors

The most important mathematical object in Riemann geometry is the **metric tensor** *g _{ij}*:
the distance between two infinitely close points in Riemann space is:

Tensor

The contravariant tensor

where

where

Using the tensors *g ^{ij}* and

Scalar multiplication can be defined in such a way:

The most fruitful idea of Riemann geometry is a special way of differentiating mathematical objects.
The components of vectors or tensors in Riemann space are changed during the parralel translations,
because the space is not linear. So the generalized derivative of vector can be written as:

*DA ^{i}/dx^{l}=dA^{i}/dx^{l} + C^{i}_{kl}A^{k}*

where

A derivative of tensor with two indexes is:

It's evident that derivative defined in such a way derives all the properties of simple algebraic derivative, for example:

Let's prove that differential of metric tensor is equal to zero. Differential

Comparing the first and the last statements it's easy to guess that

Finally, after swapping indexes and adding equations with different indexes, we obtain such an expression for

Here is a useful expression for transformations of

It's evident now that

Here are some useful expressions for vector operators in Riemann space:

- divergence:
*A*^{i}_{;i}=(g^{-1/2})d(g^{1/2}*A^{i})/dx^{i} - Laplas' operator (Laplacian):
*f*^{;i}_{;i}=(g^{-1/2})d(g^{1/2}*g^{ik}df/dx^{k})/dx^{i}

Let's define the curvature of Riemann space. We'll consider a parralel translation of some vector
*A _{i}*over a small contour and find out it's change. Then remember Stock's theorem
(see rotor/curl). Finally

where

where

Here are another useful expression for

Some properties of the curvature tensor are:

*R*^{i}_{klm}=-R^{i}_{kml}*R*^{i}_{klm}+ R^{i}_{mkl}+ R^{i}_{lmk}= 0*R*^{n}_{ikl;m}+ R^{n}_{imk;l}+ R^{n}_{ilm;k}= 0

From curvature tensor with four indexes we can obtain simplified curvature tensor with two indexes:

*R _{ik}=R^{l}_{ilk}*

Here is another expression for

This tensor is simmetrical:

Scalar curvature of the space is defined as:

Here is the Delphi unit Riemann.pas specially designed for analysing the properties of Riemann space. You
just have to define metric tensors *g _{ij}* and

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