arXiv subjects
Publications and source records attributed to Hemangi Shah.
In the present paper, we investigate Busemann functions in a general Finsler setting as well as in asymptotically harmonic Finsler manifolds. In particular, we show Busemann functions are smooth on asymptotically harmonic Finsler manifolds.
In the present paper, we introduce and investigate various types of harmonic Finsler manifolds and find out the interrelation between them. We give some characterizations of such spaces in terms of the mean curvature of geodesic spheres and the Laplacian of the distance function induced by the Finsler structure. We investigate some properties of the Finsler mean curvature of geodesic spheres of different radii. In addition, we prove that certain harmonic Finsler manifolds are of Einstein type and provide a technique to construct harmonic Finsler manifolds of Randers type. Moreover, we give some examples of non-Riemmanian Finsler harmonic manifolds of constant flag curvature and constant $S$-curvature.
$(M^n,g)$ 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.
We prove the existence of manifolds with almost maximal volume entropy which are not hyperbolic.
Let $(M,g)$ be a complete, simply connected Riemannian manifold of dimension 3 without conjugate points. We show that $M$ is a flat manifold, provided $M$ is asymptotically harmonic of constant $h = 0$.
Let (M,g) be a complete, simply connected Riemannian manifold of dimension 3 without conjugate points. We show that M is a hyperbolic manifold of constant sectional curvature, provided M is asymptotically harmonic of constant h > 0.