- #1

- 245

- 1

In Euclidean vector spaces the derivative of the position vector of a running point of a

curve is the tangent vector of the curve.

In thehttp://www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf" [Broken], on page 78 appears a vector which can be regarded as position vector in a Riemann space. This vector is the element of the tangent spce at the given point, it has the direction parallel to the direction of the geodesic connecting this point with the origin, and magnitude equal with the length of this geodesic. (Of course 'origin' means an arbitrarily fixed point of our Riemann space).

This vector is denoted by \hat \sigma on p. 78.

On this page when the author calculates the derivative of the norm of this position vector along a curve, I see that in the right side of the equation appears the tangent vector of the curve instead of the covariant derivative of the position vector.

Is it really true that in Riemann spaces also holds that the covariant derivative of the (above defined) position vector of a running point of a curve is the tangent vector of the curve? How can one see this?

curve is the tangent vector of the curve.

In thehttp://www.matematik.lu.se/matematiklu/personal/sigma/Riemann.pdf" [Broken], on page 78 appears a vector which can be regarded as position vector in a Riemann space. This vector is the element of the tangent spce at the given point, it has the direction parallel to the direction of the geodesic connecting this point with the origin, and magnitude equal with the length of this geodesic. (Of course 'origin' means an arbitrarily fixed point of our Riemann space).

This vector is denoted by \hat \sigma on p. 78.

On this page when the author calculates the derivative of the norm of this position vector along a curve, I see that in the right side of the equation appears the tangent vector of the curve instead of the covariant derivative of the position vector.

Is it really true that in Riemann spaces also holds that the covariant derivative of the (above defined) position vector of a running point of a curve is the tangent vector of the curve? How can one see this?

Last edited by a moderator: