1 Section 1 Preliminary Materials

For preliminary materials about differential geometry, we refer to the Differential Geometry I course notes.

● 1.1 Tensors and Tensor Fields

For a differential 1-form $𝜔$, we may regard it as a function accepting one vector field as the input: $\mathrm{\Gamma }\left(𝑇𝑀\right)\to {\mathrm{𝒞︀}}^{\infty }\left(𝑀\right)$, $𝑋\mapsto 𝜔\left(𝑋\right)$. Similarly, a vector field $𝑋$ is a function accepting one 1-form as the input: $𝜔\mapsto 𝜔\left(𝑋\right)$. They are both ${\mathrm{𝒞︀}}^{\infty }$-linear: meaning that, for any $𝑓\in {\mathrm{𝒞︀}}^{\infty }\left(𝑀\right)$, $𝜔\left(𝑓𝑋\right)=𝑓𝜔\left(𝑋\right)$. They are examples of tensor fields.

Given a vector space $𝑉$, a $\left(𝑘,𝑙\right)$-tensor is a multilinear map $𝑇:{\left({𝑉}^{\ast }\right)}^{\otimes 𝑘}\otimes {𝑉}^{\otimes 𝑙}\to ℝ$. $𝑘$ is called the contravariant index, the number of 1-form inputs, or the number of vector fields in the tensor; $𝑙$ is called the covariant inex, the number of vector field inputs, or the number of 1-forms in the tensor. If we let $𝑉={𝑇}_{𝑝}𝑀$ and ${𝑉}^{\ast }={𝑇}_{𝑝}𝑀$, we denote ${𝑇}_{𝑝}^{\left(𝑘,𝑙\right)}𝑀$ the space of all $\left(𝑘,𝑙\right)$-tensors on ${𝑇}_{𝑝}𝑀$. ${𝑇}^{\left(𝑘,𝑙\right)}𝑀$ is then a smooth manifold, the tensor bundle, and a smooth section is called a tensor field.

In local coordinates, a $\left(𝑘,𝑙\right)$-tensor field can be written in the form

We’re using Einstein summation convention here.

● 1.2 Lie Derivative

Lie derivatives are directional derivatives along a vector field.

Let $𝑋$ be a vector field. For a smooth function $𝑓$, the Lie derivative is defined by ${\mathrm{ℒ︀}}_{𝑋}𝑓=𝑋𝑓$. Lie derivatives of tensors are defined inductively through the Leibniz rule:

2 Section 2 Riemannian Manifolds

● 2.1 Riemannian Metric

In local coordinates, $𝑔$ can be written as $𝑔={𝑔}_{𝑖𝑗}d{𝑥}^{𝑖}d{𝑥}^{𝑗}$; since $𝑔$ is symmetric we omit the $\otimes$ symbol. $𝑔$ transforms under different coordinates as

A diffeomorphism $\mathrm{\Phi }:\left({𝑀}_1,{𝑔}_1\right)\to \left({𝑀}_2,{𝑔}_2\right)$ is called conformal, if ${\mathrm{\Phi }}^{\ast }{𝑔}_2={\mathrm{e}}^{𝑓}{𝑔}_1$ for some $𝑓\in {\mathrm{𝒞︀}}^{\infty }\left({𝑀}_1\right)$.

● 2.2 Length, Angle and Volume

Suppose $\left({𝑀}_1,{𝑔}_1\right)$ and $\left({𝑀}_2,{𝑔}_2\right)$ are Riemannian manifolds. If there is a diffeomorphism $\mathrm{\Phi }:𝑀-1\to {𝑀}_2$ such that ${\mathrm{\Phi }}^{\ast }\left({𝑔}_2\right)={𝑔}_1$, call ${𝑀}_1$ and ${𝑀}_2$ isometric, and $\mathrm{\Phi }$ an isometry. Equivalently, ${𝑔}_1\left(𝑋,𝑌\right)={𝑔}_2\left({\mathrm{\Phi }}_{\ast }𝑋,{\mathrm{\Phi }}_{\ast }𝑌\right)$.

Using the metric we can define the length of a curve. Let $𝛾:\left(𝑎,𝑏\right)\to 𝑀$ be a smooth (or ${\mathrm{𝒞︀}}^1$) curve, define its length

where $\left|\dot{𝛾}\left(𝑡\right)\right|=\sqrt{𝑔\left(\dot{𝛾}\left(𝑡\right),\dot{𝛾}\left(𝑡\right)\right)}$. For $𝑝,𝑞\in 𝑀$, we can then define the distance between $𝑝$ and $𝑞$, by

If $𝑀$ is orientable, under an oriented atlas, we can also define a volume form

Given two smooth curves ${𝛾}_1$, ${𝛾}_2$ on $𝑀$ such that they intersect at ${𝛾}_1\left({𝑡}_1\right)={𝛾}_2\left({𝑡}_2\right)=𝑝$, we define the angle $𝜃$ between them at $𝑝$ as

A diffeomorphism $\mathrm{\Phi }:\left({𝑀}_1,{𝑔}_1\right)\to \left({𝑀}_2,{𝑔}_2\right)$ is called conformal if it preserves all angles. Equivalently, $\mathrm{\Phi }$ is conformal if there is a smooth function $𝑓\in {\mathrm{𝒞︀}}^{\infty }\left({𝑀}_1\right)$ such that ${\mathrm{\Phi }}^{\ast }\left({𝑔}_2\right)={\mathrm{e}}^{𝑓}{𝑔}_1$.

● 2.3 Riemannian Metric on Products

Suppose $\left(𝑀,𝑔\right)$ and $\left(𝑁,ℎ\right)$ are two Riemannian manifolds. We give the product manifold $𝑀\times 𝑁$ a metric, the product metric $𝑔\oplus ℎ$, by

In local coordinates,

The product metric can be generalized to warped product metrics, denoted $𝑔{\oplus }_{𝑓}ℎ$, where $𝑓\in {\mathrm{𝒞︀}}^{\infty }\left(𝑀\right)$, defined as

The round metric on ${𝕊}^2$ in spherical coordinates form is an example of the warped product of two $ℝ$ with Euclidean metric.

● 2.4 Musical Isomorphisms

The Riemannian metric $𝑔$ provides a symmetric positive definite bilinear map $\mathrm{\Gamma }\left(𝑇𝑀\right)\times \mathrm{\Gamma }\left(𝑇𝑀\right)\to {\mathrm{𝒞︀}}^{\infty }\left(𝑀\right)$. This can also be viewed as a linear isomorphism $\mathrm{\Gamma }\left(𝑇𝑀\right)\to \mathrm{\Gamma }\left({𝑇}^{\ast }𝑀\right)$, given by $𝑣\mapsto 𝑔\left(𝑣,-\right)$. In other words to every vector $𝑣\in {𝑇}_{𝑝}𝑀$, there is a covector $𝐺\left(𝑣\right)$ such that $𝐺\left(𝑣\right)\left(𝑤\right)=𝑔\left(𝑣,𝑤\right)$; this $𝐺\left(𝑣\right)$ is said to be obtained from $𝑣$ by lowering the index, and is denoted ${𝑣}^{♭}$. Similarly, for a covector $𝛼$ there is a corresponding vector ${𝐺}^{-1}\left(𝑣\right)$, obtained by raising the index it is denoted ${𝛼}^{♯}$.

A remark on the names “lowering / raising the indices”. Normally, for a tensor, covariant indices are written as subscripts, and contravariant indices are written as superscripts. The name “covariant” and “contravariant” comes from the way they transform under a coordinate change: covariant tensors like 1-forms transforms by the Jacobian of the transformation

and contravariant tensors like vector fields transforms by the inverse

Therefore, turning a (contravariant) vector into a covector makes it covariant, hence lowering its index from a superscript into a subscript; and turning a covector into a vector raises its index.

By raising the index, we can define a metric $\widetilde{𝑔}$ on $\mathrm{\Gamma }\left({𝑇}^{\ast }𝑀\right)\times \mathrm{\Gamma }\left({𝑇}^{\ast }𝑀\right)$, by $\widetilde{𝑔}\left(𝛼,𝛽\right)=𝑔\left({𝛼}^{♯},{𝛽}^{♯}\right)$. In local coordinates,

where $𝛼={𝛼}_{𝑖}d{𝑥}^{𝑖}$, $𝛽={𝛽}_{𝑗}d{𝑥}^{𝑗}$ and ${𝑔}^{𝑖𝑗}$ is the inverse of ${𝑔}_{𝑖𝑗}$, ${𝑔}^{𝑖𝑘}{𝑔}_{𝑘𝑗}={𝛿}_{𝑗}^{𝑖}=\{\begin{array}{c}1\text{if}𝑖=𝑗\\ 0\text{otherwise}\end{array}$.

Similarly, we can define $𝑔$ on any $\left(𝑟,𝑠\right)$-tensor field:

3 Section 3 Parallel Transport

● 3.1 Affine Connections

Affine connections $\nabla$ are a class of “directional derivatives”. As all derivatives, we begin the definition with

On vector fields, they are characterized by the following conditions:

If $\left({𝑥}^1,\cdots ,{𝑥}^{𝑛}\right)$ is a local coordinates, $𝑉={𝑉}^{𝑖}\frac{\mathrm{\partial }}{\mathrm{\partial }{𝑥}^{𝑖}}$, $𝑊={𝑊}^{𝑖}\frac{\mathrm{\partial }}{\mathrm{\partial }{𝑥}^{𝑖}}$, then

Suppose that ${\nabla }_{\frac{\mathrm{\partial }}{\mathrm{\partial }{𝑥}^{𝑖}}}\frac{\mathrm{\partial }}{\mathrm{\partial }{𝑥}^{𝑗}}={\mathrm{\Gamma }}_{𝑖𝑗}^{𝑘}\frac{\mathrm{\partial }}{\mathrm{\partial }{𝑥}^{𝑘}}$; the ${\mathrm{\Gamma }}_{𝑖𝑗}^{𝑘}$ are called Christoffel symbols. Moreover denote

which is the usual Euclidean derivative plus a correction term. This is called the tensor component¹. Then ${\nabla }_{𝑉}𝑊$ can be written locally as ${𝑉}^{𝑖}\left({\nabla }_{𝑖}{𝑊}^{𝑗}\right)\frac{\mathrm{\partial }}{\mathrm{\partial }{𝑥}^{𝑗}}$.

If ${\nabla }_1$ and ${\nabla }_2$ are two affine connections, then ${\nabla }_1-{\nabla }_2$ is ${\mathrm{𝒞︀}}^{\infty }$-linear in both variables, so it’s a tensor field. This means that we can translate a connection with a tensor field to modify it.

The connection can also be defined on 1-forms using the Leibniz rule:

This definition is also $ℝ$-linear in $𝛼$, ${\mathrm{𝒞︀}}^{\infty }$-linear in $𝑋$, and satisfies the Leibniz rule. The tensor component is now

Generally, for $\left(𝑟,𝑠\right)$-tensor fields,

where the tensor component is

The link of affine connections to Riemannian geometry is the following theorem:

● 3.2 Parallel Transport and Holonomy

Throughout this section, let $\nabla$ be the Levi-Civita connection.

In local coordinates, $𝑉$ is parallel to $𝛾$ if and only if $\frac{d{𝑉}^{𝑘}}{d𝑡}+{\mathrm{\Gamma }}_{𝑖𝑗}^{𝑘}{𝛾}^{\prime 𝑖}\left(𝑡\right){𝑉}^{𝑗}=0$, where ${\mathrm{\Gamma }}_{𝑖𝑗}^{𝑘}$ can be calculated using Equation 29. This is a first-order linear ODE, therefore given an initial vector $𝑣$ at $𝛾\left(𝑎\right)$, we can find a unique vector field $𝑉$ on $𝛾$ such that $𝑉\left(𝑎\right)=𝑣$. In this case, we say that $𝑉\left(𝑏\right)$ is obtained from $𝑉\left(𝑎\right)$ by parallel transporting along $𝛾$. Parallel transport along $𝛾$ is denoted ${𝑃}_{𝛾}$.

Suppose $𝛾$ is a curve from $𝑝$ to $𝑞$. Parallel transport is then a map from ${𝑇}_{𝑝}𝑀$ to ${𝑇}_{𝑞}𝑀$. Moreover, it is a linear isometry, meaning that it is a linear map preserving the metric.

In particular, we can construct an orthonormal basis $\left\{{𝐸}_1,\cdots ,{𝐸}_{𝑛}\right\}$ of ${𝑇}_{𝑝}𝑀$ and parallel transport it along $𝛾$; it will remain orthonormal at any point. We thus obtain an orthonormal frame along the curve, $\left\{{𝐸}_1\left(𝑡\right),\cdots ,{𝐸}_{𝑛}\left(𝑡\right)\right\}$. If $𝑉$ is a vector field along the curve, then using this orthonormal frame, $\frac{d𝑉}{d𝑡}=\frac{d{𝑉}^{𝑖}}{d𝑡}{𝐸}_{𝑖}$.

Suppose $𝛼$ is a loop at $𝑝$. Then ${𝑃}_{𝛼}$ is a linear isometry from ${𝑇}_{𝑝}𝑀$ to ${𝑇}_{𝑝}𝑀$ itself. The set of parallel transport $\left\{{𝑃}_{𝛼}:𝛼\text{is a loop at}𝑝\right\}$ then form a subgroup of $\text{O}\left(𝑛\right)$ under composition; this is called the holonomy group ${\text{Hol}}_{𝑝}\left(𝑀,𝑔\right)$. It is a Lie subgroup of $\text{O}\left(𝑛\right)$.

If we only consider contractible loops, then ${\text{Hol}}_{𝑝}^0\left(𝑀,𝑔\right)=\left\{{𝑃}_{𝛼}:𝛼\text{is contractible}\right\}$ is a connected component of ${\text{Hol}}_{𝑝}\left(𝑀,𝑔\right)$, and ${\text{Hol}}_{𝑝}/{\text{Hol}}_{𝑝}^0$ is at most countable.

The holonomy of product is the product of holonomy:

Holonomy groups at different points are related by conjugation: if $𝛼$ is a curve from $𝑝$ to $𝑞$, then ${\text{Hol}}_{𝑝}\left(𝑀,𝑔\right)={𝑃}_{𝛼}^{-1}\cdot {\text{Hol}}_{𝑞}\left(𝑀,𝑔\right)\cdot {𝑃}_{𝛼}$.

A theorem by Ambrose-Singer says that the Lie algebra of ${\text{Hol}}_{𝑝}^0$ is generated by curvature forms.

${\text{Hol}}_{𝑝}^0$ has a natural action on ${𝑇}_{𝑝}𝑀$. Decompose ${𝑇}_{𝑝}𝑀$ into irreducible invariant subspaces ${𝑉}_1\oplus \cdots \oplus {𝑉}_{𝑘}$. Along parallel transport, the above decomposition preserves. Therefore, along a curve $𝛾$, ${𝑇}_{𝛾}𝑀$ can be decomposed into distributions ${𝐷}_1\oplus \cdots \oplus {𝐷}_{𝑘}$.

If a Riemannian manifold cannot be decomposed into a product, it is called irreducible.

4 Section 4 Geodesics

Locally, $\left({𝑥}_{𝑖}\left(𝑡\right)\right)$ is a geodesic if and only if

for all $𝑖$. This is a (quasi-linear) ODE, so it has a unique solution under two initial conditions: $𝛾\left(0\right)=𝑝$ and ${𝛾}^{\prime }\left(0\right)=𝑣\in {𝑇}_{𝑝}𝑀$, and this unique solution is smooth.

If every maximal geodesic is defined on $ℝ$, or if every geodesic can be indefinitely extended, call this manifold geodesic complete.

● 4.1 Exponential map

Suppose $𝛾:\left[0,1\right]\to 𝑀$ is a geodesic with $𝛾\left(0\right)=𝑝$ and ${𝛾}^{\prime }\left(0\right)=𝑣$. We define the exponential map ${exp}_{𝑝}:{𝑇}_{𝑝}𝑀\to 𝑀$ as ${exp}_{𝑝}\left(𝑣\right)=𝛾\left(1\right)$.

If $𝑀$ is not geodesically complete, the exponential map may not be defined everywhere on ${𝑇}_{𝑝}𝑀$. The domain of ${exp}_{𝑝}$ is always a star-shaped region (because ${exp}_{𝑝}\left(𝜆𝑣\right)=𝛾\left(𝜆\right)$ for $𝜆<1$). We often restrict it to a small ball around $0\in {𝑇}_{𝑝}𝑀$.

In this case we call ${exp}_{𝑝}\left({𝐵}_{𝜀\left(0\right)}\right)$ a geodesic ball and its boundary ${exp}_{𝑝}\left({𝑆}_{𝜀}\left(0\right)\right)$ a geodesic sphere. $𝑉\subseteq 𝑀$ is called a normal neighbourhood if there is some $𝑈\in {𝑇}_{𝑝}𝑀$ such that ${exp}_{𝑝}:𝑈\to 𝑉$ is a diffeomorphism.

${exp}_{𝑝}$ induces two sets of local coordinates around $𝑝$. The first is the normal coordinate: it’s the image of the orthonormal Cartesian coordinates on ${𝑇}_{𝑝}𝑀$ on a normal neighbourhood. Under normal coordinates, ${\mathrm{\Gamma }}_{𝑖𝑗}^{𝑘}\left(𝑝\right)=0$ and $𝑔\left({𝑒}_{𝑖},{𝑒}_{𝑗}\right)={𝛿}_{𝑖𝑗}$ at $𝑝$. (Notice: it’s only valid at $𝑝$, and not around $𝑝$!)

The second is the geodesic polar coordinates. It is the image of polar coordinates on ${𝑇}_{𝑝}𝑀$ on a normal neighbourhood. Therefore the coordinates consists of radial geodesics from $𝑝$ and geodesic spheres around $𝑝$. Marvelously, orthogonality of polar coordinates are preserved.

● 4.2 Hopf-Rinow Theorem

As a corollary, all compact manifolds are complete.

● 4.3 Length Minimizers: First Variation of Arclength

Suppose $𝛾:\left[𝑎,𝑏\right]\to 𝑀$ is a (piecewise) smooth curve. A variation family $\mathrm{\Phi }$ of $𝛾$ is map $\mathrm{\Phi }:\left[𝑎,𝑏\right]\times \left(-𝜀,𝜀\right)\to 𝑀$ such tht

1. $\mathrm{\Phi }\left(-,0\right)=𝛾$;
2. $\mathrm{\Phi }$ is (piecewise in the first parameter) smooth.

We asy that $\mathrm{\Phi }$ is proper if $\mathrm{\Phi }\left(𝑎,-\right)=𝛾\left(𝑎\right)$ and $\mathrm{\Phi }\left(𝑏,-\right)=𝛾\left(𝑏\right)$.

As a corollary, if $\mathrm{\Phi }$ is a proper variation and ${𝑣}_0$ is chosen arbitrarily, then $𝛾$ is arclength-minimizing only if $𝛾$ is a geodesic.

5 Section 5 Curvature

● 5.1 Hessian and Laplacian

If the Levi-Civita connection corresponds to the first order derivative, then the Hessian corresponds to second order derivative.

Recall that ${\nabla }_{𝑋}𝑓=𝑋\left(𝑓\right)$ can be treated as a covector field: $\nabla 𝑓:𝑋\mapsto {\nabla }_{𝑋}𝑓$. By lowering the index, we get the gradient vector field

$\nabla \nabla 𝑓$ is then a $\left(2,0\right)$-tensor field: $\nabla \nabla 𝑓:\left(𝑋,𝑌\right)\mapsto \left({\nabla }_{𝑋}\left(\nabla 𝑓\right)\right)\left(𝑌\right)$. By some calculation, $\nabla \nabla 𝑓=𝑋\left(𝑌\left(𝑓\right)\right)-\left({\nabla }_{𝑋}𝑌\right)\left(𝑓\right)$. This is called the Hessian. Locally:

Therefore, the Hessian is symmetric in $𝑋$ and $𝑌$. Conventionally we write $\nabla \nabla 𝑓={\nabla }_{𝑖}{\nabla }_{𝑗}𝑓\cdot d{𝑥}^{𝑖}d{𝑥}^{𝑗}$, where

Similarly, $\nabla 𝑌:𝑋\mapsto {\nabla }_{𝑋}𝑌$ is a $\left(1,1\right)$-tensor field. Locally,

The Laplacian of a function $𝑓$ is defined as $∆𝑓=tr\nabla \left(\text{grad}𝑓\right)$. Locally,

The last step is by metric compatibility. Actually, we have the following formula:

● 5.2 Riemannian Curvature

For vector fields we can do similar things. Let $𝑋$, $𝑌$ and $𝑍$ be vector fields. Then

Therefore,

Remarkably, the first term, which is equal to ${\nabla }_{𝑋}{\nabla }_{𝑌}𝑍-{\nabla }_{𝑌}{\nabla }_{𝑋}𝑍-{\nabla }_{\left[𝑋,𝑌\right]}𝑍$, is a tensor. This is the Riemannian curvature tensor $𝑅\left(𝑋,𝑌\right)𝑍$. This is a $\left(1,3\right)$-tensor field, and it can be written as $𝑅={𝑅}_{𝑖𝑗𝑘}^{𝑙}d{𝑥}^{𝑖}\otimes d{𝑥}^{𝑗}\otimes \frac{\mathrm{\partial }}{\mathrm{\partial }𝑙}\otimes {𝑥}^{𝑘}$. The $𝑖$, $𝑗$, $𝑘$ indices corresponds to the $𝑋$, $𝑌$, $𝑍$ input, and the $𝑙$ index corresponds to the field $𝑅\left(𝑋,𝑌\right)𝑍$.

Expanding the defining equation into Christoffel symbols, we get

By lowering the index $𝑙$ we get a $\left(0,4\right)$-tensor field

In local coordinates, ${𝑅}_{𝑖𝑗𝑘𝑙}^{}={𝑔}_{𝑘𝑝}^{}{𝑅}_{𝑖𝑗𝑙}^{𝑝}$.

$𝑅\left(𝑋,𝑌\right):𝑍\mapsto 𝑅\left(𝑋,𝑌\right)𝑍$ is called the curvature operator. Locally, $𝑅\left({\mathrm{\partial }}_{𝑖},{\mathrm{\partial }}_{𝑗}\right)𝑍=\left({𝑍}^{𝑙}{𝑅}_{𝑖𝑗𝑙}^{𝑘}\right){\mathrm{\partial }}_{𝑙}$, so

Extending $𝑅\left(𝑋,𝑌\right)$ to ${\mathrm{\Omega }}^1\left(𝑀\right)$,

For functions, $𝑅\left(𝑋,𝑌\right)𝑓=0$, because the Hessian is symmetric.

By anti-symmetric properties of ${𝑅}_{𝑚}$, it is actually a map ${\wedge }^2𝑇𝑀\otimes {\wedge }^2𝑇𝑀\to {\mathrm{𝒞︀}}^{\infty }\left(𝑀\right)$, sending $\left(𝑋\wedge 𝑌,𝑍\wedge 𝑊\right)$ to ${𝑅}_{𝑚}\left(𝑋,𝑌,𝑍,𝑊\right)$. Moreover, we can define an operator $𝑅:{\wedge }^2𝑇𝑀\to {\wedge }^2𝑇𝑀$, such that $⟨𝑅\left(𝑋\wedge 𝑌\right),𝑍\wedge 𝑊⟩={𝑅}_{𝑚}\left(𝑋\wedge 𝑌,𝑍\wedge 𝑊\right)$. This $𝑅$ is self-adjoint, and is called the curvature operator.

● 5.3 Sectional Curvature, Ricci Curvature, Scalar Curvature

In dimension $2$, there can be only one tangent plane at every point, and the sectional curvature is equal to the Gaussian curvature: $𝐾=\frac{{𝑅}_{1212}}{{\Vert {\mathrm{\partial }}_1\times {\mathrm{\partial }}_2\Vert }^2}$.

The sectional curvature can be defined for any $\left(0,4\right)$-tensor satisfying the symmetry properties in Proposition 5.1. Conversely, if for two such $\left(0,4\right)$-tensors ${𝑅}_{𝑚}$ and ${\widetilde{𝑅}}_{𝑚}$, we get the same sectional curvature at $𝑝$ for all $𝜎$ (${𝐾}_{𝑝}\left(𝜎\right)={\widetilde{𝐾}}_{𝑝}\left(𝜎\right)$), then ${𝑅}_{𝑚}\left(𝑝\right)={\widetilde{𝑅}}_{𝑚}\left(𝑝\right)$.

If for any $𝑝$ and any $𝜎$, ${𝐾}_{𝑝}\left(𝜎\right)$ is constant, we call $𝑀$ a manifold of constant sectional curvature. If $𝑀$ has constant curvature $𝑐\in ℝ$, then

If $𝑀$ is a simply connected manifold of constant curvature, then $𝑀$ must be isometric to a sphere, the Euclidean space, or a hyperbolic space.

The geometric meaning of this identity is encoded in the Hilbert-Einstein functional $\text{HE}\left(𝑔\right)=\int {𝑆}_{𝑔}d{\text{vol}}_{𝑔}$. By the second Bianchi identity, this is diffeomorphism invariant.

● 5.4 The Laplacian Operator

We introduce some analysis-flavoured results on manifolds. The key is the following expansion of the Laplacian:

Similar formulas also exist for Laplacians of (harmonic) 1-forms, etc.

The typical use of this formula is in certain special cases, for example, when $\text{Ric}$ is positive / negative definite, when $𝑓$ is a harmonic function (so that $\nabla \left(∆𝑓\right)=𝜆\nabla 𝑓$), etc. The following theorem is useful in working with harmonic functions:

Therefore, for example, to prove a function on a compact manifold is constant, one only need to prove that it’s subharmonic, and we can use Bochner’s formula to estimate the Laplacian.

Some examples of statements that can be proved using these results:

6 Section 6 Jacobi Fields and Second Variation

Jacobi fields arise in variational families of geodesics. Formally, if $𝐹\left(𝑠,𝑡\right)={𝛾}_{𝑠}\left(𝑡\right):\left(-𝜀,𝜀\right)\times \left[0,𝑎\right]\to 𝑀$ is a variational family of geodesics, i.e. ${𝛾}_{𝑠}$ is a geodesic for all $𝑠$, then the variational vector field $𝐽\left(𝑡\right)={\frac{\mathrm{\partial }𝐹}{\mathrm{\partial }𝑠}|}_{𝑠=0}$ is a Jacobi field. Conversely, if $𝐽\left(𝑡\right)$ is a Jacobi field along a geodesic $𝛾$, then there is a variational family of geodesics $𝐹\left(𝑠,𝑡\right)={𝛾}_{𝑠}\left(𝑡\right)$ such that $𝐽\left(𝑡\right)={\frac{\mathrm{\partial }𝐹}{\mathrm{\partial }𝑠}|}_{𝑠=0}$. The construction can be visualized by the following diagram.

Choosing a parallel frame $\left\{{𝐸}_{𝑖}\right\}$ along $𝛾$ and set $𝐽\left(𝑡\right)={𝑓}^{𝑖}\left(𝑡\right){𝐸}_{𝑖}\left(𝑡\right)$, then the equation may be written as a system of second order linear ODEs:

Let $𝛾$ be a geodesic. $𝐽\left(𝑡\right)=\dot{𝛾}\left(𝑡\right)$ is a Jacobi field, representing parallel translation of $𝛾$ along itself, ${𝛾}_{𝑠}\left(𝑡\right)=𝛾\left(𝑡+𝑠\right)$. $𝐽\left(𝑡\right)=𝑡\dot{𝛾}\left(𝑡\right)$ is also a Jacobi field, representing a linear reparametrization of gamma, ${𝛾}_{𝑠}\left(𝑡\right)=𝛾\left(\left(1+𝑠\right)𝑡\right)$. By linearity of the Jacobi field equation, $\left(𝑎+𝑏𝑡\right)\dot{𝛾}\left(𝑡\right)$ are all Jacobi fields. In fact, they are the only parallel Jacobi fields.

Therefore, all Jacobi fields can be decomposed as ${𝐽}^{⟂}\left(𝑡\right)+\left(𝑎+𝑏𝑡\right)\dot{𝛾}\left(𝑡\right)$, where ${𝐽}^{⟂}\left(𝑡\right)$ is orthogonal to $\dot{𝛾}\left(𝑡\right)$; such a Jacobi field is called a normal Jacobi field.

Suppose $𝐽\left(0\right)=0$. Then Taylor expanding ${\left|𝐽\left(𝑡\right)\right|}^2$ about $𝑡$ gives

where $𝑣={𝛾}^{\prime }\left(0\right)$ and $𝑤={𝐽}^{\prime }\left(0\right)$. Therefore, in negative sectional curvature, $𝐽\left(𝑡\right)$ is “longer” than in the Euclidean case, and in positive sectional curvature, $𝐽\left(𝑡\right)$ is “shorter” than in the Euclidean case (think hyperbolic planes and spheres). Similarly, in normal coordinates,

● 6.1 Conjugate Loci

Suppose $𝛾:\left[0,𝑎\right]\to 𝑀$ is a geodesic and ${𝑡}_0\in \left[0,𝑎\right]$. If there is a Jacobi field along $𝛾$ vanishing at both $𝛾\left(0\right)$ and $𝛾\left({𝑡}_0\right)$, call $𝛾\left({𝑡}_0\right)$ a conjugate point of, or conjugate to, $𝛾\left(0\right)$. The number of linearly independent such Jacobi fields is called the multiplicity of this conjugate point.

If ${𝛾}^{\prime }\left(0\right)=𝑣$, then $𝛾\left({𝑡}_0\right)$ is conjugate to $𝛾\left(0\right)$ if and only if ${𝑡}_0𝑣$ is a critical point of ${\left({exp}_{𝑝}\right)}_{\ast }$, i.e. there is some $𝑤$ such that ${\left({exp}_{𝑝}\right)}_{\ast ,{𝑡}_0𝑣}\left(𝑤\right)=0$. If $𝛾\left({𝑡}_0\right)$ is not conjugate to $𝛾\left(0\right)$, then there is a unique Jacobi field from $0$ to ${𝑡}_0$, given arbitrary boundary conditions $𝐽\left(0\right)=𝑣$, $𝐽\left(𝑎\right)=𝑤$.

The conjugate locus of $𝑝$ is defined as

● 6.2 Second Variation of Arclength and the Index Form

If we fix the endpoints, the boundary term vanishes, and we are left with

We can make this into a bilinear form in two vector fields, by defining the index form

The index form is a symmetric bilinear form; its null space is the space of all Jacobi fields along $𝛾$ vanishing at the endpoints.

For a variational vector field $𝑉$, $𝐼\left(𝑉,𝑉\right)⩾𝐼\left({𝑉}^{⟂},{𝑉}^{⟂}\right)={\frac{d^2}{d{𝑠}^2}|}_{𝑠=0}𝐿\left({𝛾}_{𝑠}\right)$. Therefore, a normalized geodesic $𝛾$ is arclength-minimizing only if the index form along $𝛾$ is positive semi-definite.