Riemannian geometry of the space of volume preserving immersions

Abstract

Abstract. On a closed manifold of dimension greater than one, every smooth weak Riemannian metric on the space of smooth positive probability densities, that is invariant under the action of the diffeomorphism group, is a multiple of the Fisher-Rao metric.Introduction. The Fisher-Rao metric on the space Prob(M ) of probability densities is of importance in the field of information geometry. Restricted to finitedimensional submanifolds of Prob(M ), so-called statistical manifolds, it is called Fisher's information metric [1]. The Fisher-Rao metric has the property that it is invariant under the action of the diffeomorphism group. The interesting question is whether it is the unique metric possessing this invariance property. A uniqueness result was established [4, p. 156] for Fisher's information metric on finite sample spaces and [2] extended it to infinite sample spaces.

The gradient flow of the potential energy on the space of arcs

“…The celebrated catenary problem of finding the shape of a stationary chain with two fixed ends corresponds to the time-independent solution to (4). Galileo [19] thought that it was a parabola, but Johann Bernoulli, Leibniz and Huygens proved that it was a hyperbolic cosine.…”

“…cf. (4). Assume that the gravity is of the same order as the damping, that is, f g = cg for some constant vector g. Letting σ = ς/c and c → +∞ we formally deduce ∂ t η(t, s) = ∂ s (σ(t, s)∂ s η(t, s)) + g, |∂ s η(t, s)| = 1.…”

“…We refer to [20,Section 2.6] for the details and for a discussion of links between the inextensible string (4), the geodesic interpretation of motion of an ideal incompressible fluid (associated with the names of Arnold and Brenier), and the optimal transport theory. In particular, Euler's equations for ideal incompressible fluid, together with the whip equations (2), are examples of geodesic equations on infinite-dimensional manifolds of volume preserving immersions endowed with the L 2 metric [4]. Apart from that, (4) can be recast [20] into a discontinuous system of conservation laws as well as into so-called "total variation wave equation".…”

Uniformly Compressing Mean Curvature Flow

“…We first observe that the geodesics inà k are determined by the condition ∂ τ τ γ ⊥ T γÃk , which can be expressed as (91) ∂ τ τ γ = ς H(γ) + dγ · grad γ * · ς, γ ∈à k , cf. [11,34]. Then we can calculate the Hessian of the L 2 -mass, taking into account (84) (with w = γ and div γ(M) (γ • γ −1 ) = k):…”

Isometric Immersions and the Waving of Flags

“…A natural extension of this to higher dimensions is to consider the space of embeddings of surfaces into R 3 with some constraint: either preserving the area element or preserving the Riemannian metric. Those that preserve the area element, which serve as a model for the motion of membranes in biological systems, were studied by several researchers including the first author [5,15,27]. In this article we study those that preserve the metric, which can serve as a model for unstretchable fabric or paper.…”