# Geometry of Asymptotically harmonic manifolds with minimal horospheres

## TL;DR

This paper proves that simply connected, asymptotically harmonic manifolds with minimal horospheres are necessarily flat, extending previous results on harmonic manifolds and providing a broader understanding of their geometric structure.

## Contribution

The paper establishes that such manifolds are flat by demonstrating their parallelizable structure and isometric embedding into Euclidean space, generalizing prior flatness results for harmonic manifolds.

## Key findings

- The vector space spanned by Busemann functions is finite dimensional and equals the manifold's dimension.
- The gradients of Busemann functions form a global parallel orthonormal basis.
- The manifold admits an isometry to Euclidean space, confirming flatness.

## Abstract

$(M^n,g)$ be a complete Riemannian manifold without conjugate points. In this paper, we show that if $M$ is also simply connected, then $M$ is flat, provided that $M$ is also asymptotically harmonic manifold with minimal horospheres (AHM). The (first order) flatness of $M$ is shown by using the strongest criterion: $\{{e_i}\}$ be an orthonormal basis of $T_{p}M$ and $\{b_{e_{i}}\}$ be the corresponding Busemann functions on $M$. Then, (1) The vector space $V = span\{b_{v} | v \in T_{p}M \}$ is finite dimensional and dim $V = $ dim $M = n$.(2) $\{\nabla b_{e_i}(p) \}$ is a global parallel orthonormal basis of $T_{p}M$ for any $p \in M$. Thus, $M$ is a parallizable manifold. And (3) F : M -> R^n defined by $F(x) = (b_{e_1}(x), b_{e_{2}}(x), \cdots, b_{e_{n}}(x)),$ is an isometry and therefore, $M$ is flat. Consequently, AH manifolds can have either polynomial or exponential volume growth,generalizing the corresponding result of [18] for harmonic manifolds. In case of harmonic manifold with minimal horospheres (HM), the (second order) flatness was proved in [23] by showing that $span\{b_{v}^2 | v \in T_{p}M \}$ is finite dimensional. We conclude that, the results obtained in this paper are the strongest and wider in comparison to harmonic manifolds, which are known to be AH. In fact, our proof shows the more generalized result, viz.: If (M,g) is a non-compact, complete, connected Riemannian manifold of infinite injectivity radius and of subexponential volume growth, then M is a first order flat manifold.