# Totally geodesic submanifolds of Teichmuller space

## TL;DR

This paper proves that in Teichmüller space, higher-dimensional totally geodesic submanifolds are contained within finitely many totally geodesic subvarieties, revealing a finiteness property of such structures.

## Contribution

It establishes the finiteness of higher-dimensional totally geodesic subvarieties in each moduli space and shows they cover totally geodesic submanifolds.

## Key findings

- Any higher-dimensional totally geodesic submanifold covers a totally geodesic subvariety.
- Only finitely many such subvarieties exist in each moduli space.
- Higher-dimensional totally geodesic submanifolds are contained within these subvarieties.

## Abstract

We show that any totally geodesic submanifold of Teichmuller space of dimension greater than one covers a totally geodesic subvariety, and only finitely many totally geodesic subvarieties of dimension greater than one exist in each moduli space.