Isometries and isometric embeddings of Wasserstein spaces over the Heisenberg group

Abstract: Our purpose in this paper is to study isometries and isometric embeddings of the

p

-Wasserstein space

m a t h c a l W_{p} (m a t h b b H^{n})

over the Heisenberg group

m a t h b b H^{n}

for all

p > 1

and for all

n g e q 1

. First, we create a link between optimal transport maps in the Euclidean space

m a t h b b R^{2 n}

and the Heisenberg group

m a t h b b H^{n}

. Then we use this link to understand isometric embeddings of

m a t h b b R

and

m a t h b b R_{+}

into

m a t h c a l W_{p} (m a t h b b H^{n})

for

p > 1

. That is, we characterize complete geodesics and geodesic rays in the Wasserstein space. Using these results we determine the metric rank of

m a t h c a l W_{p} (m a t h b b H^{n})

. Namely, we show that

m a t h b b R^{k}

can be embedded isometrically into

m a t h c a l W_{p} (m a t h b b H^{n})

for

p > 1

if and only if

k l e q n

. As a consequence, we conclude that

m a t h c a l W_{p} (m a t h b b R^{k})

and

m a t h c a l W_{p} (m a t h b b H^{k})

can be embedded isometrically into

m a t h c a l W_{p} (m a t h b b H^{n})

if and only if

k l e q n

. In the second part of the paper, we study the isometry group of

m a t h c a l W_{p} (m a t h b b H^{n})

for

p > 1

. We find that these spaces are all isometrically rigid meaning that for every isometry

P h i : m a t h c a l W_{p} (m a t h b b H^{n}) o m a t h c a l W_{p} (m a t h b b H^{n})

there exists a

p s i : m a t h b b H^{n} o m a t h b b H^{n}

such that

.

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