# Selfsimilar Hessian manifolds

## TL;DR

This paper characterizes selfsimilar Hessian manifolds, showing their structure is either conical or Euclidean, and describes the local form of radiant selfsimilar Hessian manifolds, expanding understanding of their geometric properties.

## Contribution

It provides a comprehensive characterization of global and local selfsimilar Hessian manifolds, including their structure and classification under potential homothetic vector fields.

## Key findings

- Selfsimilar manifolds with potential homothetic vector fields are conical or Euclidean.
- Selfsimilar Hessian manifolds are locally products of radiant Hessian manifolds.
- The structure of radiant selfsimilar Hessian manifolds is explicitly described.

## Abstract

A selfsimiar manifold is a Riemannian manifold $\left(M,g\right)$ endowed with a homothetic vector field $\xi$. We characterize global selfsimilar manifolds and describe the structure of local selfsimilar manifolds. We prove that any selfsimilar manifold with a potential homothetic vector field is a conical Riemannian manifold or a Eucledean space. A radiant Hessian manifold is selfsimilar Hessian manifold $\left(M,\nabla,g,\xi\right)$ such that $\nabla\xi=\lambda \text{Id}$. We prove that any selfsimilar Hessian manifold with a potential homothetic vector field is locally isomorphic to a product radiant Hessian manifolds and describe the local structure of radiant selfsimialar Hessian manifolds.