# Metric geometry and analysis on Einstein manifolds

> **NSF 01002324DB NSF RESEARCH & RELATED ACTIVIT** · University of California-San Diego (CA) · $222,327

## Abstract

Metric Riemannian geometry is a central subject in modern mathematics. The original concept dates back to Bernhard Riemann's famous Habilitation lecture "Ueber die hypothesen, welche der Geometrie zu Grunde liegen" (On the hypotheses which lie at the bases of geometry) delivered on 10 June 1854. The revolutionary creations in this lecture profoundly changed the global landscape of geometry. Specifically, Riemann proposed a novel strategy to generalize the geometry of surfaces to higher dimensions which he called Mannigfaltigkeiten (manifolds). A large variety of new notions and concepts were created: these include the notion of curvature which quantitatively measures how a space is curved, and the notion of geodesic which is a length-minimizing path connecting two points on a manifold. The studies of the metric structures of manifolds, what we now call metric Riemannian geometry, primarily focuses on the interplay between the global geometry of the underlying space and the metric structure, namely how the distance between two points can be realized or measured. This project is mainly concerned with the metric geometry of Einstein manifolds where the metric structures satisfy the Einstein equation in the theory of general relativity.  The PI will integrate their research with training and mentorship at a variety of levels.  This includes organizing summer workshops and mathematical retreats on  Riemannian geometry; complex geometry and theoretical physics, and designing and d

## Key facts

- **NSF award ID:** 2550348
- **Awardee organization:** University of California-San Diego (CA)
- **SAM.gov UEI:** UYTTZT6G9DT1
- **PI:** Ruobing Zhang
- **Primary program:** 01002324DB NSF RESEARCH & RELATED ACTIVIT
- **All programs:** —
- **Estimated total:** $222,327
- **Funds obligated:** $74,838
- **Transaction type:** Standard Grant
- **Period:** 07/01/2025 → 07/31/2026

## Primary source

NSF Award Search: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2550348

## Citation

> US National Science Foundation, Award 2550348, Metric geometry and analysis on Einstein manifolds. Retrieved via AI Analytics 2026-06-07 from https://api.ai-analytics.org/grant/nsf/2550348. Licensed CC0.

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