AI 中文总结
本文针对弹性微分拓扑空间，利用其切函子的真实切结构发展黎曼几何，引入适配联络、协变导数等，证明测地线相关性质，给出映射空间上结构的逐点构造。
AI 中文摘要
微分拓扑空间存在多种不等价的切空间概念，这阻碍了该框架下黎曼几何的发展。针对Blohmann提出的弹性微分拓扑空间，其切函子具有Cockett和Cruttwell意义下的真实切结构，我们利用这一结构发展黎曼几何。我们引入适配该场景的联络，证明其能诱导满足常规公理的向量场协变导数及相关曲率张量，并按照Cockett-Cruttwell的方法记录曲率张量的基本性质。给定黎曼度量，我们建立Koszul公式，证明Levi-Civita协变导数若存在则唯一，但其一般存在性仍待解决。我们进一步发展测地线与能量泛函理论：测地线恒为能量泛函的临界路径；对于实直线为曲线对象的弹性空间，测地线喷雾的完备性可推出测地线的存在性与唯一性，反之临界路径即为测地线。作为基础实例，我们证明：对于闭流形M与弹性黎曼微分拓扑空间N，映射空间C^∞(M,N)上的这些结构可通过N上的结构逐点得到，当N为流形时，自然的切交换假设自动成立；特别地，当N的测地线喷雾完备时，C^∞(M,N)的测地线喷雾也完备。
英文摘要
Several inequivalent notions of tangent space exist for diffeological spaces, which has hindered the development of Riemannian geometry in this setting. On Blohmann's elastic diffeological spaces, the tangent functor carries a genuine tangent structure in the sense of Cockett and Cruttwell, and we use this to develop Riemannian geometry. We introduce connections adapted to this setting and show that they induce covariant derivatives on vector fields satisfying the usual axioms, together with the associated curvature tensor, whose basic properties we record following Cockett--Cruttwell. Given a Riemannian metric, we establish a Koszul formula and prove that the Levi-Civita covariant derivative is unique if it exists, while its existence remains open in general. We then develop the theory of geodesics and of the energy functional: geodesics are always critical paths of the energy functional, and on elastic spaces for which the real line is a curve object, completeness of the geodesic spray yields existence and uniqueness of geodesics, as well as the converse statement that critical paths are geodesics. As a fundamental class of examples, we show that for a closed manifold $M$ and an elastic Riemannian diffeological space $N$ these structures on the mapping space $C^\infty(M,N)$ are obtained pointwise from those on $N$, under a natural tangent-commuting hypothesis that holds automatically when $N$ is a manifold; in particular, the geodesic spray of $C^\infty(M,N)$ is complete whenever that of $N$ is.
Comments29 pages. v2: corrections and clarifications
