Thanks for the very informative reply. We have two situations:
1. No-torsion condition: The path independence property (Section A, in the image) of the incremental displacement vector leads to symmetry of basis vectors. The Christoffel symbol formula (Section B in the image) is not merely a mathematical definition but its derivation is based on symmetry of the incremental displacement vector, which is the property of geometry of space.
A metric gives us the incremental displacement vector and the corresponding unit vectors. The unit vectors are essential for suggesting a coordinate system corresponding to the space under study. This is required to write any vector in the given space.
By a geometrical picture, I mean a coordinate system, corresponding unit vectors and a metric describing curvature (if any) of the space.
2. Non-zero torsion: Kindly refer to the reply:
A non-zero torsion situation is described as: “A parallel transport over a closed path in a space with torsion produces a displacement from the original position, like how curvature changes a vector's orientation. On a discrete lattice, this means that torsion is equivalent to dislocation defects”.
If it is difficult to describe a metric for this non-zero torsion situation, then it will also be difficult to write the unit vectors. It will be difficult to write a coordinate system or write any vector in such space. Therefore, I desired to know the geometrical aspects such as a metric, corresponding unit vectors and a coordinate system to describe such a space. It is appropriate to derive a revised Christoffel symbols formula corresponding to such space, based on these geometrical characteristics.
3. I also wonder, if it is absolutely necessary to define a curved space (non-zero torsion) to describe a non-closing shape such as a parallelepiped?
A Cartesian coordinate system is suitable to describe a straight line or a three dimensional cube. We shift to the spherical coordinate system to study a circle or a sphere. But the space still remains flat. Similarly a non-closing parallelogram (or parallelepiped) may also be mathematically described in a flat space. May be, each side will have to be studied independently by dividing it into small increments. The analysis may be more complicated but the space need not be called curved.
Thanks once again for your reply and I am raising these comments for remarks by the experts.
Ref: Physics Overflow submission:
A classical scrutiny of the Schwarzschild solution