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# A Review of Lectures on Differential Geometry by Yau and Schoen: A Classic Book on Advanced Topics in Geometry

## Lectures on Differential Geometry by Yau and Schoen: A Review

If you are interested in learning about differential geometry, one of the most fascinating and important branches of mathematics, you might want to check out the book Lectures on Differential Geometry by Shing-Tung Yau and Richard Schoen. This book is based on a series of lectures given by the authors at Harvard University in 1980, and it covers some of the most advanced topics in the field. In this article, we will give you a brief overview of what differential geometry is, who Yau and Schoen are, and what are the main topics covered in the book. We will also highlight some of the key results and insights that the authors present in each chapter.

## Introduction

### What is differential geometry?

Differential geometry is the study of geometric shapes and structures using tools from calculus, linear algebra, and topology. It deals with questions such as: How can we measure distances, angles, areas, volumes, curvatures, etc. on curved surfaces or spaces? How can we compare different shapes or spaces using these measurements? How can we deform or transform one shape or space into another while preserving some properties? How can we use geometric ideas to describe physical phenomena such as gravity, light, heat, etc.?

Differential geometry has many applications in science, engineering, art, and philosophy. For example, it is essential for understanding general relativity, which is the theory of gravity proposed by Albert Einstein. It also plays a role in computer graphics, computer vision, robotics, cryptography, string theory, cosmology, etc.

### Who are Yau and Schoen?

Shing-Tung Yau and Richard Schoen are two of the most influential and distinguished mathematicians in the world. They have made many groundbreaking contributions to differential geometry and related fields such as partial differential equations, complex analysis, algebraic geometry, etc. They have received numerous awards and honors for their work, including the Fields Medal (the highest prize in mathematics), the Wolf Prize (the second highest prize in mathematics), the Crafoord Prize (the equivalent of the Nobel Prize for mathematics), the MacArthur Fellowship (the "genius grant"), etc.

Yau was born in China in 1949 and moved to Hong Kong when he was eight years old. He obtained his Ph.D. from the University of California at Berkeley in 1971 under the supervision of Shiing-Shen Chern, who is considered to be the founder of modern differential geometry. He then became a professor at Stanford University, where he met Schoen. He later moved to Harvard University, where he is currently a professor emeritus. He is also a professor at Tsinghua University in Beijing.

Schoen was born in Ohio in 1950. He obtained his Ph.D. from Stanford University in 1977 under the supervision of Leon Simon, who is a leading expert in geometric analysis. He then became a professor at the University of California at Berkeley, where he collaborated with Yau on several projects. He later moved to Stanford University, where he is currently a professor emeritus. He is also a professor at the University of California at Irvine.

### What are the main topics covered in the book?

The book Lectures on Differential Geometry consists of four chapters, each of which covers a major theme in differential geometry. The chapters are:

• Chapter 1: Riemannian Geometry

• Chapter 2: Complex Manifolds

• Chapter 3: Harmonic Maps

• Chapter 4: Positive Mass Theorem and Black Holes

In each chapter, the authors introduce the basic concepts and definitions, state and prove some of the most important theorems and results, and give some examples and applications. The book is intended for advanced undergraduate or graduate students who have some background in calculus, linear algebra, and topology. It is also suitable for researchers who want to learn more about the topics covered in the book.

## Chapter 1: Riemannian Geometry

### Basic concepts and definitions

Riemannian geometry is the study of curved spaces that are endowed with a notion of distance or metric. A metric is a function that assigns a positive number to any pair of points in a space, such that it satisfies some properties such as symmetry, triangle inequality, etc. A metric allows us to measure lengths of curves, angles between vectors, areas of surfaces, volumes of regions, etc.

A Riemannian manifold is a space that locally looks like a Euclidean space (a flat space) with a metric. For example, the surface of a sphere is a Riemannian manifold, because every small patch on the sphere can be flattened onto a plane with a metric. A Riemannian manifold can be described by giving a set of coordinates and a matrix of functions that define the metric in each coordinate system.

### Curvature and its properties

Curvature is a measure of how much a Riemannian manifold deviates from being flat. There are different ways to define curvature, depending on what aspect of the geometry we want to capture. Some of the most common types of curvature are:

• Gaussian curvature: This is the product of the principal curvatures at each point on a surface. It measures how much the surface bends in different directions. For example, a sphere has positive Gaussian curvature everywhere, because it curves away from any tangent plane. A cylinder has zero Gaussian curvature everywhere, because it curves only in one direction. A saddle has negative Gaussian curvature everywhere, because it curves towards opposite sides of any tangent plane.

• Ricci curvature: This is the trace of the sectional curvature at each point on a manifold. It measures how much the volume of a small ball around each point changes when it is transported along different directions. For example, a sphere has positive Ricci curvature everywhere, because any small ball shrinks when it is moved along any direction. A flat space has zero Ricci curvature everywhere, because any small ball stays the same when it is moved along any direction. A hyperbolic space has negative Ricci curvature everywhere, because any small ball expands when it is moved along any direction.

• Scalar curvature: This is the sum of the Ricci curvature at each point on a manifold. It measures how much the volume of a small ball around each point changes when it is scaled by a factor. For example, a sphere has positive scalar curvature everywhere, because any small ball becomes smaller when it is scaled down by any factor. A flat space has zero scalar curvature everywhere, because any small ball stays the same when it is scaled by any factor. A hyperbolic space has negative scalar curvature everywhere, because any small ball becomes larger when it is scaled down by any factor.

Curvature has many important properties and implications for the geometry and topology of Riemannian manifolds. For example:

• The Gauss-Bonnet theorem states that the integral of the Gaussian curvature over a compact surface equals to 2π times its Euler characteristic (a topological invariant that depends only on the number of holes in the surface).

• The Ricci flow is a process that deforms a Riemannian manifold by smoothing out its Ricci curvature. It was used by Grigori Perelman to prove the Poincaré conjecture (a famous problem that asks whether every simply connected three-dimensional manifold is homeomorphic to a sphere).

### Comparison theorems and applications

Comparison theorems are results that compare the geometry of a Riemannian manifold with that of a simpler model space, such as a sphere, a flat space, or a hyperbolic space. They allow us to deduce global properties of a manifold from its local curvature. For example:

• The Cartan-Hadamard theorem states that if a complete Riemannian manifold has nonpositive sectional curvature everywhere, then it is simply connected (has no holes) and its universal cover (a space that covers it without overlaps) is isometric to a Euclidean space.

• The Bonnet-Myers theorem states that if a complete Riemannian manifold has positive Ricci curvature everywhere, then it has finite volume and diameter (the maximum distance between any two points).

• The Cheeger-Gromoll theorem states that if a complete Riemannian manifold has nonnegative Ricci curvature everywhere, then it has a finite number of ends (parts that look like infinite cylinders) and each end has a nonnegative scalar curvature.

Comparison theorems have many applications in geometry and analysis. For example, they can be used to prove the existence and uniqueness of geodesics (shortest curves) between any two points on a manifold, to estimate the eigenvalues and eigenfunctions of the Laplace operator (a differential operator that measures how much a function varies on a manifold), to study the heat equation (a partial differential equation that describes how heat flows on a manifold), etc.

## Chapter 2: Complex Manifolds

### Complex structures and holomorphic maps

A complex manifold is a space that locally looks like a complex Euclidean space (a flat space with complex coordinates) with a metric. A complex structure is a way of assigning complex coordinates to each point on a manifold, such that the transition functions between different coordinate systems are holomorphic (complex differentiable). A holomorphic map is a function between two complex manifolds that preserves the complex structure.

Complex manifolds are generalizations of complex curves (one-dimensional complex manifolds) and complex surfaces (two-dimensional complex manifolds). They have many beautiful properties and symmetries that make them easier to study than general Riemannian manifolds. For example:

• The Cauchy-Riemann equations relate the partial derivatives of a holomorphic function with respect to the real and imaginary parts of the complex coordinates. They imply that holomorphic functions are infinitely differentiable and satisfy the mean value property (the value of a function at any point is equal to the average value over any small disk around that point).

• The Hodge decomposition theorem states that any differential form (a function that assigns numbers or vectors to small patches on a manifold) on a compact complex manifold can be uniquely decomposed into four types: harmonic (satisfies the Laplace equation), exact (the differential of another form), coexact (the negative of the differential of another form), and primitive (orthogonal to all harmonic forms).

• The Dolbeault cohomology groups measure the failure of exactness of holomorphic forms (differential forms that are holomorphic in each coordinate system) on a complex manifold. They are analogous to the de Rham cohomology groups, which measure the failure of exactness of smooth forms on a smooth manifold.

### Kähler metrics and Ricci curvature

A Kähler metric is a special type of metric on a complex manifold that is compatible with both the complex structure and the symplectic structure (a way of measuring areas and volumes on a manifold). A Kähler manifold is a complex manifold that admits a Kähler metric. A Kähler metric can be described by giving a potential function, called the Kähler potential, from which the metric can be derived.

Kähler manifolds are important examples of Riemannian manifolds that have many nice properties and structures. For example:

• The Levi-Civita connection (a way of transporting vectors along curves on a manifold) on a Kähler manifold is compatible with both the complex structure and the symplectic structure. This implies that parallel transport (moving vectors along curves while keeping them parallel) preserves angles, lengths, areas, volumes, etc.

• The Ricci curvature tensor (a way of measuring how much the volume element changes under parallel transport) on a Kähler manifold is determined by the Kähler potential. This implies that the Ricci curvature is invariant under biholomorphic transformations (holomorphic maps that have holomorphic inverses).

• The Calabi conjecture states that any Kähler manifold with a given Ricci curvature can be obtained by deforming the Kähler potential of another Kähler manifold. This conjecture was proved by Yau in 1978, using a nonlinear partial differential equation called the Monge-Ampère equation.

### Calabi-Yau manifolds and mirror symmetry

A Calabi-Yau manifold is a special type of Kähler manifold that has zero Ricci curvature everywhere. A Calabi-Yau manifold has a trivial canonical bundle (a bundle of differential forms that measures the orientation of the manifold) and a finite number of holomorphic forms (differential forms that are holomorphic in each coordinate system). A Calabi-Yau manifold can be described by giving a complex algebraic variety (a set of solutions to a system of polynomial equations) and a complex structure on it.

Calabi-Yau manifolds are important examples of Riemannian manifolds that have many applications in physics and mathematics. For example:

• In string theory, a theory that attempts to unify quantum mechanics and general relativity, Calabi-Yau manifolds are used to model the extra dimensions of space-time that are not observable in our four-dimensional world.

• In mirror symmetry, a conjecture that relates two different types of Calabi-Yau manifolds, called mirror pairs, Calabi-Yau manifolds are used to study the geometry and topology of algebraic varieties and their moduli spaces (spaces that parametrize all possible shapes of the varieties).

• In enumerative geometry, a branch of mathematics that counts the number of geometric objects that satisfy certain conditions, Calabi-Yau manifolds are used to compute the Gromov-Witten invariants (numbers that count the number of curves on a manifold) and the Donaldson-Thomas invariants (numbers that count the number of sheaves on a manifold).

## Chapter 3: Harmonic Maps

### Harmonic functions and harmonic forms

A harmonic function is a function on a Riemannian manifold that satisfies the Laplace equation, which is a partial differential equation that states that the divergence of the gradient of the function is zero. A harmonic function minimizes the Dirichlet energy, which is a functional that measures how much the function varies on the manifold. A harmonic function is also characterized by the mean value property, which states that the value of the function at any point is equal to the average value over any small ball around that point.

A harmonic form is a differential form on a Riemannian manifold that satisfies the Hodge equation, which is a partial differential equation that states that the Laplace operator applied to the form is zero. A harmonic form minimizes the Hodge energy, which is a functional that measures how much the form varies on the manifold. A harmonic form is also characterized by being closed (the differential of the form is zero) and co-closed (the negative of the differential of the dual form is zero).

Harmonic functions and harmonic forms are important examples of solutions to elliptic partial differential equations, which are equations that have no real characteristic curves (curves along which solutions can propagate). They have many applications in physics and mathematics. For example:

• In potential theory, a branch of mathematics that studies functions that arise from physical phenomena such as gravity, electrostatics, heat, etc., harmonic functions are used to model potentials (functions that determine forces or energies) on Riemannian manifolds.

• In Hodge theory, a branch of mathematics that studies cohomology groups (groups that measure the failure of exactness of differential forms) on Riemannian manifolds, harmonic forms are used to construct canonical representatives (forms that minimize energy) for each cohomology class (equivalence class of forms).

• In de Rham theory, a branch of mathematics that relates cohomology groups (groups that measure the failure of exactness of differential forms) on smooth manifolds with homology groups (groups that measure the number of holes in manifolds), harmonic forms are used to construct Poincaré duals (forms that correspond to submanifolds) for each homology class (equivalence class of submanifolds).

### Energy and stability of harmonic maps

the manifolds. A harmonic map satisfies the Euler-Lagrange equation, which is a partial differential equation that states that the divergence of the differential of the map is zero. A harmonic map is also characterized by being a critical point of the energy functional, which means that the variation of the energy functional along any direction is zero.

The energy and stability of harmonic maps are important topics in geometric analysis, which is a branch of mathematics that studies geometric problems using analytical methods. For example:

• The Dirichlet problem for harmonic maps asks whether there exists a harmonic map between two Riemannian manifolds that agrees with a given map on the boundary of the domain manifold. This problem is related to the existence and uniqueness of solutions to elliptic partial differential equations with boundary conditions.

• The regularity theory for harmonic maps studies the smoothness and singularity of harmonic maps between Riemannian manifolds. This theory is related to the elliptic estimates and the removable singularity theorem for elliptic partial differential equations.

• The stability analysis for harmonic maps investigates the behavior of harmonic maps under small perturbations or deformations. This analysis is related to the second variation and the index theorem for elliptic partial differential equations.

### Minimal surfaces and constant mean curvature surfaces

A minimal surface is a surface in a Riemannian manifold that minimizes the area functional, which is a functional that measures the area of the surface. A minimal surface satisfies the minimal surface equation, which is a partial differential equation that states that the mean curvature of the surface is zero. A minimal surface is also characterized by being a critical point of the area functional, which means that the variation of the area functional along any direction is zero.

A constant mean curvature surface is a surface in a Riemannian manifold that has constant mean curvature everywhere. A constant mean curvature surface satisfies the constant mean curvature equation, which is a partial differential equation that states that the Laplacian of the position vector of the surface is proportional to the mean curvature vector of the surface. A constant mean curvature surface is also characterized by being an extremal point of the area functional, which means that the second variation of the area functional along any direction is nonnegative or nonpositive.

Minimal surfaces and constant mean curvature surfaces are important examples of harmonic maps, because they can be viewed as maps from a two-dimensional domain manifold to a higher-dimensional target manifold that minimize or extremize some energy functional. They have many applications in physics and mathematics. For example:

In soap film theory, a branch of physics that studies physical phenomena involving thin films of liquid, minimal surfaces are used to model soap films (surfaces that minimize surface tension) and soap bubbles (surfaces that enclose a fixed volume with minimal surfac