The quadratic distance function on a Riemannian manifold can be expressed in terms of the position vector, which in turn can be constructed using geodesic normal coordinates through consideration of the exponential map. The formulas for the derivative of the distance are useful to study Lyapunov stability of dynamical systems, and to build cost functions for optimal control and estimation.Index Terms-Riemannian geometry, geodesic distance, Lyapunov functions.
I. NORMAL COORDINATES AND THE POSITION VECTORConsider an n-dimensional manifold X on which we can define a Riemannian metric , , and pick a particular point O as the origin of a system of geodesic normal coordinates. These are obtained by considering all geodesic curves that depart from O. Each geodesic has some initial velocity V , that is, it is a curve γ(t), t ∈ [0, 1] such that(1)Writing the components of the vector V at the origin as {x i }, we then assign the coordinates {x i } to the point γ(1).Repeating the construction for each V in some subset of T O X containing the null vector, we obtain a local system of coordinates around O, which may be extended to any region which does not contain points conjugate to O, that is, where the geodesic curves do not cross or touch. The classic construction of the map from T O X to X thus defined, called the exponential map, is detailed in any good book on Riemannian geometry, such as [1]. With the family of geodesics so defined, we may call the vector field X taking the values X| γ(1) =γ(1) the "position vector." Its expression in the said normal coordinates is X = x i ∂/∂x i . (In contrast, the expression in normal coordinates of the radial geodesic passing through a fixed point {x i } at t = 1 is γ i (t) = tx i .) Keep in mind however than the definition of X is intrinsic, that is to say, independent of the coordinates chosen.In the following sections we shall describe the quadratic distance function and its properties in terms of the vector field X, show how these properties can be useful in the study F Pait is grateful for the generous support of Susanna Stern and the continued interest of readers like you. He is with the Escola Politécnica (which unlike its French counterpart is not called X), of Lyapunov stability, and establish related formulas for the Lie derivatives of the metric tensor and the connection. These formulas will be used in future work concerning algorithms for the "recursive" construction of geometries with appropriate properties.
II. PROPERTIES OF THE QUADRATIC DISTANCE FUNCTIONConsider the quadratic distance function defined at each point γ(1) of a manifold bywhere γ(0) = O,γ i (0) = x i , and ∇γγ = 0, that is, γ(t) is a geodesic connecting O and a point γ(1) whose normal coordinates are {x i }. In this paper, we will always consider ∇ as the torsion-free, Levi-Civita connection compatible with the metric , . Functional (2), the integral of the square norm of a curve, is variously called the action integral by physicists and the energy of the curve by mathematicians. We prefer to think of ...
