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I must tell you that I have never seen this kind of question in Tensor Analysis. Our professor had set up this question in our exam, but I don't know whether it belongs to tensors or not. The question goes like thiS:

1. If a vector $u^{i}$ gets parallel transported along a curve S, then $$u^{i}{}_{; j}~ \frac{dX^{j}}{dS}~=~0.$$ 2. If the angle between $u^{i}$ and $v^{j}$ is $\theta$, show that getting parallel transported does not change angle between them.

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  • $\begingroup$ Your notation seems a bit non-standard to me, especially the part with the $\mathrm{d}X/\mathrm{d}S$. Is it a vector field? Please elaborate a bit on your question. $\endgroup$
    – ACuriousMind
    Commented Jul 18, 2014 at 17:21
  • $\begingroup$ That has really been the problem for me as well,first of all, is there any mistakes in my question...Even I am struggling to understand the question $\endgroup$ Commented Jul 18, 2014 at 17:27
  • $\begingroup$ Also, with respect to which connection do we parallel transport (Levi-Civita naturally comes to mind)? $\endgroup$
    – ACuriousMind
    Commented Jul 18, 2014 at 17:29
  • $\begingroup$ I want to rephrase this question as- Show that the angle between two vectors remain invariant or does not change in parallel displacement parallel transport. $\endgroup$ Commented Jul 19, 2014 at 0:59

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A vector $u^a$ is parallel-transported along the integral curve of tangent vector $V^a \equiv \frac{dX^a}{d s}$ if we have $V^a\nabla_a u^b =0$ (a vector parallel-transported with respect to itself defines a geodesic). The angle between two vectors $v^a$ and $u^a$ is given by $\cos\theta = \frac{v^a u_a}{\sqrt{v^av_a u^bu_b}}$, hence we take: $$ V^a \nabla_a (u^b v_b) = v_bV^a \nabla_a u^b+ V^a u^b \nabla v_b = 0,$$ since the vectors $v^a$ and $u^a$ are parallel-transported with respect to $V^a$. Since $V^a \nabla_a (u^b u_b) = 0$ for the same reason, $\theta$ remains constant along the integral curve of $V^a$.

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  • $\begingroup$ I am sorry, but I am pretty much lost in the symbols being used in here, could you provide me the references or links to study it further on my own. $\endgroup$ Commented Jul 21, 2014 at 23:08
  • $\begingroup$ I'm using $\nabla_a$ instead of ${}_{;a}$ and $s$ is the affine parameter of $X^a$. $\endgroup$
    – auxsvr
    Commented Jul 21, 2014 at 23:11
  • $\begingroup$ Thanks, any references or links for me to further explore since I am not the student of GR and want to sincerely learn about this from basic though I have sound knowledge on tensors, well at least in basic level $\endgroup$ Commented Jul 21, 2014 at 23:14
  • $\begingroup$ If you use google, there are lots of lecture notes on GR, but they require understanding of calculus, linear algebra, mechanics, differential geometry, differential equations etc. $\endgroup$
    – auxsvr
    Commented Jul 21, 2014 at 23:34

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