1 geodesics

A curve on a surface that locally minimizes distance

A geodesic in an n-surface S in \mathbb R^{n+1} is a parametrized curve \alpha: I \to S whose acceleration is orthogonal to S at every point, \forall t \in I, \ddot \alpha(t) \in S_{\alpha(t)}^\perp. Geodesic has constant speed.

Theorem:

Let S be an n-surface, let p \in S and let v \in S_p. Then there exists an open interval I containing 0 and a geodesic \alpha: I \to S such that:

1. \alpha(0) = p and \dot \alpha(0) = v
2. \alpha is maximal

1.1 vector field along a parametrized curve

• A vector field \vec X along a parametrized curve \alpha: I \to \mathbb R^{n+1} is a function which assigns to each t \in I a vector \vec X(t) at \alpha(t). \vec X(t) = (\alpha(t), X_1(t), \cdots, X_{n+1}(t))
  • \frac{d}{dt} (\vec X + \vec Y) = \frac{d\vec X}{dt} + \frac{d\vec Y}{dt}
  • \frac{d}{dt} (f\vec X) = \frac{df}{dt}\vec X + f\frac{d\vec X}{dt}
  • \frac{d}{dt} (\vec X \cdot \vec Y) = \frac{d \vec X}{dt}\cdot\vec Y + \vec X\cdot\frac{d \vec Y}{dt}
• A function f along \alpha is simply a function f: I \to \mathbb R

1.2 parallel transport

1.2.1 covariant derivative

A tangent vector field is obtained by projecting \dot X orthogonally to S_{\alpha(t)}. The covariant derivative measures the rate of change of X along \alpha as registered on S.

X' = \dot X - [\dot X \cdot N(\alpha(t))]N(\alpha(t))

1.2.1.1 properties

• (X + Y)' = X' + Y'
• (fX)' = f'X + fX'
• (X \cdot Y)' = X' \cdot Y + X \cdot Y'
• A parametrized curve \alpha is a geodesics in S iff its covariant acceleration (\dot \alpha)’ is zero along \alpha

1.2.2 euclidean parallel

Let \vec v = (p, v) \in \mathbb R_{p}^{n+1} and \vec w = (q, w) \in \mathbb R_{q}^{n+1}

A vector field \vec X is Euclidean parallel along \alpha if X(t_1) = X(t_2) for all t_1, t_2 \in I

1.2.3 Levi-Civita parallel

A smooth vector field X tangent to S along \alpha is called Levi-Civita parallel if X'(t) = 0 for all t \in I.

1.2.3.1 properties

• If X is parallel along \alpha then X has constant length
• If X and Y are parallel along \alpha then X \cdot Y is constant along \alpha
• If X and Y are parallel along \alpha then the angle between them is constant
• If X and Y are parallel along \alpha then so are X + Y and cX for c \in \mathbb R
• The velocity vector field along \alpha in S is parallel iff \alpha is a geodesic

1.2.3.2 theorems

1.2.3.2.1 1

There exists a unique vector field V tangent to S along \alpha which is parallel and has V(t_0 \in I) = v \in S_{\alpha(t_0)}

1.2.3.2.2 2

Each parametrized curve \alpha: [a; b] \to S from p to q determines a map P_\alpha: S_p \to S_q given by P_\alpha(v) = V(b). Where V is the unique parallel vector field along \alpha with V(a) = v. The vector P_\alpha(v) is called the parallel transport of v from p to q along \alpha.