Now I'm reading a book Einstein Gravity in a Nutshell by A. Zee, and I had some misunderstanding.
Zee writes (page 89):

before that, he wrote,

Trying to solve the problem. I expanded in a series metrics as mentioned [itex]g_{\theta\theta}(r) = r^2_*+2r_*(r-r_*)+2(r-r_*)^2[/itex]
I see that [itex]B_{\mu\nu, \lambda \sigma}=B_{\theta\theta,r r}=B_{22,11}=1[/itex].

How can I get intrinsic curvature?
Below, he still writes

I turns out that the curvature is not zero, it should be.

To expand on this some more: "locally flat coordinates" means Cartesian coordinates, i.e., you are supposed to set up a local chart centered on the point (r*, 0) in which the metric at that point is ##ds^2 = dx^2 + dy^2##. So the metric coefficients you should be expanding to 2nd order are really not ##g_{rr}## and ##g_{\theta \theta}## anyway; they are ##g_{xx}##, ##g_{yy}##, and ##g_{xy} = g_{yx}##. You know that ##g_{xx} (0) = g_{yy} (0) = 1## and ##g_{xy} (0) = g_{yx} (0) = 0## since the (x, y) chart is locally flat at its origin. So in order to expand out the first and second-order terms for the locally flat metric coefficients, you just need to know the coordinate transformation from the global ##(r, \theta)## coordinates to the local ##(x, y)## coordinates, so you can transform the line element in general--i.e., not just at the single point (r*, 0).

I think you must go on to the next page. Normally the curvature would depend on both the A's and B's. Since the expression you gave involves B's only, Zee must expect you to transform the coordinates further to eliminate the A's.

Well, put the problem in another way. Let us forget for a while about Zee. How to find locally-flat coordinates (x,y) on the sphere at a north pole, say? What it means to find? It is clear that they are Cartesian, so I need to find transition formulas.

From what you've said, I still don't know just what Zee is planning to do. What he's calling "locally flat coordinates" are more usually called quasi-Cartesian coordinates or normal coordinates at a given point P. To find them, one can

a) Choose a set of orthonormal vectors X_{i} at P.
b) Solve the geodesic equations to construct a set of geodesics starting at P with initial tangent X_{i}.
c) Use these geodesics as a set of coordinate axes in the neighborhood of P, defining the normal coordinates as distance measured along each axis.

Just a roundabout way of finding coordinates in which the Christoffel symbols vanish at P, which is what Zee's A coefficients are.