Second-Order Calculus on RCD Spaces

Abstract

In this paper we develop a general 'analytic' splitting principle for RCD spaces: we show that if there is a function with suitable Laplacian and Hessian, then the space is (isomorphic to) a warped product.Our result covers most of the splitting-like results currently available in the literature about RCD spaces. We then apply it to extend to the non-smooth category some structural property of Riemannian manifolds obtained by Li and Wang.

On the notion of parallel transport on RCD spaces

“…2 Some basic notions To keep the presentation short, we shall assume the reader familiar with the language proposed in [8] (see also [6]).…”

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We propose a general notion of parallel transport on RCD spaces, prove an unconditioned uniqueness result and existence under suitable assumptions on the space.

MSC2010: 30Lxx, 51Fxx2 Some basic notions To keep the presentation short, we shall assume the reader familiar with the language proposed in [8] (see also [6]).

Curves in Banach spacesWe recall here some basic results about measurability and integration of Banach-valued maps of a single variable t ∈ [0, 1].

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Recognizing the flat torus among $$\mathsf{RCD}^*(0,N)$$ RCD ∗ ( 0 , N )

“…To keep the presentation at reasonable length we shall assume the reader familiar with the notions of Sobolev functions (see [10], [28], [4]), of differential calculus on metric measure spaces (see [15], [13]) and with the notion of Regular Lagrangian Flow on metric measure spaces ( [8], [9]). Here we shall only recall a few facts mainly to fix the notation.…”

“…a) Using the approximation property (2.2.2) one can show that S 2 loc (X) can be replaced with either one of S 2 (X), W 1,2 (X) in the above statement b) If one chooses to replace S 2 loc (X) with either S 2 (X) of W 1,2 (X), then it is also possible to replace the L 0 -normed module with a L 2 -normed module in the statement and in this case in (ii) 'L 0 -linear' should be replaced by 'L ∞ -linear' (notice that the choice of the module also affects the topology considered, whence the possibility of having two different uniqueness results). The proof is unaltered: compare for instance Theorem 3.2 with the construction of pullback module given in [15] and [13]. c) Call (L 2 (T * X), d) the outcome of Theorem 2.4 written for L 2 -normed modulus and one of the spaces S 2 (X), W 1,2 (X).…”

“…The space of vector fields L 0 (T X) is defined as the dual of the L 0 -normed module L 0 (T * X). Equivalently, it is the L 0 -completion of the dual L 2 (T X) of the L 2 -normed module L 2 (T * X) (see [15], [13]). L 2 loc (T X) ⊂ L 0 (T X) is the space of X's such that |X| ∈ L 2 loc (X).…”