微分几何笔记讨论班的粗略总结Chp7-2

（回到目录）

Chp7. Riemmannian Geometry

7.4.2

The projection method is just like in the 3-D geometry, at both endpoints, we can draw two lines that are vertical to the tangent plane. There exits a unique connection which has vanishing torsion。
7.4.3

In the example 7.4, we note the metric is diagonal and the elements are constant. After we write down the metric, we can solve the curve equations by geodesic equations.
x' is a tangent vector, in eq.(7.58).
Eq.(7.59) is just as same as the Euler-Lagrange motion equation in the QFT. It's a variational problem. The method is same. So we compute the geodesic equation, then the Christoffel symbols(Levi-Civita connection) can be calculated.
7.4.4 The normal coordinate system

In this section, p is a initial point of a vector. Point q is in the neighbourhood of point p, so the tangent space of q is just the tangent space of p(p is near q). Then the vector pq is in the tangent space of point p, obviously \phi(p)=0 that is the vector pp=0. We note c(0)=p, c(1)=q, from \phi(q)=X_q^{\mu}, we can know \phi(r)=X_q^{\mu}t, where the point r is between p and q. Obviously the paramter t give a linear function, so introducing the normal coordinate system is a technology of linearizing space.
And now, we have a symmetric connection that implys the Riemann curvature tensor has a simple form.


相关附件地址rar+pdf格式

（回到目录）