Notes on Riemannian Geometry

Recently I have been studying differential geometry, including Riemannian geometry. When studying this subject, a lot of aha moments came up due to my previous (albeit informal) exposure to the geometric point-of-view of natural gradient method. I found that the argument from this point-of-view to be very elegant, which motivates me further to study geometry in depth. This writing is a collection of small notes (largely from Lee’s Introduction to Smooth Manifolds and Introduction to Riemannian Manifolds) that I find useful as a reference on this subject. Note that, this is by no means a completed article. I will update it as I study further.

Manifolds

We are interested in generalizing the notion of Euclidean space into arbitrary smooth curved space, called smooth manifold. Intuitively speaking, a topological $n$-manifold $\M$ is a topological space that locally resembles $\R^n$. A smooth $n$-manifold is a topological $n$-manifold equipped with locally smooth map $\phi_p: \M \to \R^n$ around each point $p \in \M$, called the local coordinate chart.

Example 1 (Euclidean spaces). For each $n \in \mathbb{N}$, the Euclidean space $\R^n$ is a smooth $n$-manifold with a single chart $\phi := \text{Id}_{\R^n}$, the identity map, for all $p \in \M$. Thus, $\phi$ is a global coordinate chart.

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Example 2 (Spaces of matrices). Let $\text{M}(m \times n, \R)$ denote the set of $m \times n$ matrices with real entries. We can identify it with $\R^{mn}$ and as before, this is a smooth $mn$-dimensional manifold. Some of its subsets, e.g. the general linear group $\text{GL}(n, \R)$ and the space of full rank matrices, are smooth manifolds.

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Remark 1. We will drop $n$ when referring a smooth $n$-manifold from now on, for brevity sake. Furthermore, we will start to use the Einstein summation convention: repeated indexes above and below are implied to be summed, e.g. $v_i w^i := \sum_i v_i w^i$.

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Tangent vectors and covectors

At each point $p \in \M$, there exists a vector space $T_p \M$, called the tangent space of $p$. An element $v \in T_p \M$ is called the tangent vector. Let $f: \M \to \R$ be a smooth function. In local coordinate $\{x^1, \dots, x^n\}$ defined around $p$, the coordinate vectors $\{ \partial/\partial x^1, \dots, \partial/\partial x^n \}$ form a coordinate basis for $T_p \M$.

A tangent vector $v \in T_p \M$ can also be seen as a derivation, a linear map $C^\infty(\M) \to \R$ that follows Leibniz rule (product rule of derivative), i.e.

$$v(fg) = f(p)vg + g(p)vf \enspace \enspace \forall f, g \in C^\infty(\M) \, .$$

Thus, we can also see $T_p \M$ to be the set of all derivations of $C^\infty(\M)$ at $p$.

For each $p \in \M$ there also exists the dual space $T_p^* \M$ of $T_p \M$, called the cotangent space at $p$. Each element $\omega \in T_p^* \M$ is called the tangent covector, which is a linear functional $\omega: T_p \M \to \R$ acting on tangent vectors at $p$. Given the same local coordinate as above, the basis for the cotangent space at $p$ is called the dual coordinate basis and is given by $\{ dx^1, \dots, dx^n \}$, such that $dx^i(\partial/\partial x^j) = \delta^i_j$ the Kronecker delta. Note that, this implies that if $v := v^i \, \partial/\partial x^i$, then $dx^i(v) = v^i$.

Tangent vectors and covectors follow different transformation rules. We call an object with lower index, e.g. the components of tangent covector $\omega_i$ and the coordinate basis $\partial/\partial x^i =: \partial_i$, to be following the covariant transformation rule. Meanwhile an object with upper index, e.g. the components a tangent vector $v^i$ and the dual coordinate basis $dx^i$, to be following the contravariant transformation rule. These stem from how an object transform w.r.t. change of coordinate. Recall that a vector, when all the basis vectors are scaled up by a factor of $k$, the coefficients in its linear combination will be scaled by $1/k$, thus it is said that a vector transforms contra-variantly (the opposite way to the basis). Analogously, we can show that when we apply the same transformation to the dual basis, the covectors coefficients will be scaled by $k$, thus it transforms the same way to the basis (co-variantly).

The partial derivatives of a scalar field (real valued function) on $\M$ can be interpreted as the components of a covector field in a coordinate-independent way. Let $f$ be such scalar field. We define a covector field $df: \M \to T^* \M$, called the differential of $f$, by

$$df_p(v) := vf \enspace \enspace \text{for} \, v \in T_p\M \, .$$

Concretely, in smooth coordinates $\{ x^i \}$ around $p$, we can show that it can be written as

$$df_p := \frac{\partial f}{\partial x^i} (p) \, dx^i \, \vert_p \, ,$$

or as an equation between covector fields instead of covectors:

$$df := \frac{\partial f}{\partial x^i} \, dx^i \, .$$

The disjoint union of the tangent spaces at all points of $\M$ is called the tangent bundle of $\M$

$$TM := \coprod_{p \in \M} T_p \M \, .$$

Meanwhile, analogously for the cotangent spaces, we define the cotangent bundle of $\M$ as

$$T^*M := \coprod_{p \in \M} T^*_p \M \, .$$

If $\M$ and $\mathcal{N}$ are smooth manifolds and $F: \M \to \mathcal{N}$ is a smooth map, for each $p \in \M$ we define a map

$$dF_p : T_p \M \to T_{F(p)} \mathcal{N} \, ,$$

called the differential of $F$ at $p$, as follows. Given $v \in T_p \M$:

$$dF_p (v)(f) := v(f \circ F) \, .$$

Moreover, for any $v \in T_p \M$, we call $dF_p (v)$ the pushforward of $v$ by $F$ at $p$. It differs from the previous definition of differential in the sense that this map is a linear map between tangent spaces of two manifolds. Furthermore the differential of $F$ can be seen as the generalization of the total derivative in Euclidean spaces, in which $dF_p$ is represented by the Jacobian matrix.

Vector fields

If $\M$ is a smooth $n$-manifold, a vector field on $\M$ is a continuous map $X: \M \to T\M$, written as $p \mapsto X_p$, such that $X_p \in T_p \M$ for each $p \in \M$. If $(U, (x^i))$ is any smooth chart for $\M$, we write the value of $X$ at any $p \in U \subset \M$ as

$$X_p = X^i(p) \, \frac{\partial}{\partial x^i} \vert_p \, .$$

This defines $n$ functions $X^i: U \to \R$, called the component functions of $X$. The restriction of $X$ to $U$ is a smooth vector field if and only if its component functions w.r.t. the chart are smooth.

Example 3 (Coordinate vector fields). If $(U, (x^i))$ is any smooth chart on $\M$, then $p \mapsto \partial/\partial x^i \vert_p$ is a vector field on $U$, called the i-th coordinate vector field. It is smooth as its component functions are constant. This vector fields defines a basis of the tangent space at each point.

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Example 4 (Gradient). If $f \in C^\infty(\M)$ is a real-valued function on $\M$, then the gradient of $f$ is a vector field on $\M$. See the corresponding section below for more detail.

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We denote $\mathfrak{X}(\M)$ to be the set of all smooth vector fields on $\M$. It is a vector space under pointwise addition and scalar multiplication, i.e. $(aX + bY)_p = aX_p + bY_p$. The zero element is the zero vector field, whose value is $0 \in T_p \M$ for all $p \in \M$. If $f \in C^\infty(\M)$ and $X \in \mathfrak{X}(\M)$, then we define $fX: \M \to T\M$ by $(fX)_p = f(p)X_p$. Note that this defines a multiplication of a vector field with a smooth real-valued function. Furthermore, if in addition, $g \in C^\infty(\M)$ and $Y \in \mathfrak{X}(\M)$, then $fX + gY$ is also a smooth vector field.

A local frame for $\M$ is an ordered $n$-tuple of vector fields $(E_1, \dots, E_n)$ defined on an open subset $U \subseteq M$ that is linearly independent and spans the tangent bundle, i.e. $(E_1 \vert_p, \dots, E_n \vert_p)$ form a basis for $T_p \M$ for each $p \in \M$. It is called a global frame if $U = M$, and a smooth frame if each $E_i$ is smooth.

If $X \in \mathfrak{X}(\M)$ and $f \in C^\infty(U)$, we define $Xf: U \to \R$ by $(Xf)(p) = X_p f$. $X$ also defines a map $C^\infty(\M) \to C^\infty(\M)$ by $f \mapsto Xf$ which is linear and Leibniz, thus it is a derivation. Moreover, derivations of $C^\infty(\M)$ can be identified with smooth vector fields, i.e. $D: C^\infty(\M) \to C^\infty(\M)$ is a derivation if and only if it is of the form $Df = Xf$ for some $X \in \mathfrak{X}(\M)$.

Tensors

Let $\{ V_k \}$ and $U$ be real vector spaces. A map $F: V_1 \times \dots \times V_k \to U$ is said to be multilinear if it is linear as a function of each variable separately when the others are held fixed. That is, it is a generalization of the familiar linear and bilinear maps. Furthermore, we write the vector space of all multilinear maps $ V_1 \times \dots \times V_k \to U $ as $ \text{L}(V_1, \dots, V_k; U) $.

Example 5 (Multilinear functions). Some examples of familiar multilinear functions are

1. The dot product in $ \R^n $ is a scalar-valued bilinear function of two vectors. E.g. for any $ v, w \in \R^n $, the dot product between them is $ v \cdot w := \sum_i^n v^i w^i $, which is linear on both $ v $ and $ w $.
2. The determinant is a real-valued multilinear function of $ n $ vectors in $ \R^n $.

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Let $\{ W_l \}$ also be real vector spaces and suppose

$$\begin{align} F&: V_1 \times \dots \times V_k \to \R \\ G&: W_1 \times \dots \times W_l \to \R \end{align}$$

be multilinear maps. Define a function

$$\begin{align} F \otimes G &: V_1 \times \dots \times V_k \times W_1 \times \dots \times W_l \to \R \\ F \otimes G &(v_1, \dots, v_k, w_1, \dots, w_k) = F(v_1, \dots, v_k) G(w_1, \dots, w_l) \, . \end{align}$$

From the multilinearity of $ F $ and $ G $ it follows that $ F \otimes G $ is also multilinear, and is called the tensor product of $ F $ and $ G $. I.e. tensors and tensor products are multilinear map with codomain in $ \R $.

Example 6 (Tensor products of covectors). Let $ V $ be a vector space and $ \omega, \eta \in V^* $. Recall that they both a linear map from $ V $ to $ \R $. Therefore the tensor product between them is

$$\begin{align} \omega \otimes \eta &: V \times V \to \R \\ \omega \otimes \eta &(v_1, v_2) = \omega(v_1) \eta(v_2) \, . \end{align}$$

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Example 7 (Tensor products of dual basis). Let $ \epsilon^1, \epsilon^2 $ be the standard dual basis for $ (\R^2)^* $. Then, the tensor product $ \epsilon^1 \otimes \epsilon^2: \R^2 \times \R^2 \to \R $ is the bilinear function defined by

$$\epsilon^1 \otimes \epsilon^2(x, y) = \epsilon^1 \otimes \epsilon^2((w, x), (y, z)) := wz \, .$$

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We use the notation $ V_1^* \otimes \dots \otimes V_k^* $ to denote the space $ \text{L}(V_1, \dots, V_k; \R) $. Let $ V $ be a finite-dimensional vector space. If $ k \in \mathbb{N} $, a covariant $ k $-tensor on $ V $ is an element of the $ k $-fold tensor product $ V^* \otimes \dots \otimes V^* $, which is a real-valued multilinear function of $ k $ elements of $ V $ to $ \R $. The number $ k $ is called the rank of the tensor.

Analogously, we define a contravariant $ k $-tensor on $ V $ to be an element of the element of the $ k $-fold tensor product $ V \otimes \dots \otimes V $. We can mixed the two types of tensors together: For any $ k, l \in \mathbb{N} $, we define a mixed tensor on $ V $ of type $ (k, l) $ to be the tensor product of $ k $ such $ V $ and $ l $ such $ V^* $.

Riemannian metrics

So far we have no mechanism to measure the length of (tangent) vectors like we do in standard Euclidean geometry, where the length of a vector $v$ is measured in term of the dot product $ \sqrt{v \cdot v} $. Thus, we would like to add a structure that enables us to do just that to our smooth manifold $\M$.

A Riemannian metric $ g $ on $ \M $ is a smooth symmetric covariant 2-tensor field on $ \M $ that is positive definite at each point. Furthermore, for each $ p \in \M $, $ g_p $ defines an inner product on $ T_p \M $, written $ \inner{v, w}_g = g_p(v, w) $ for all $ v, w \in T_p \M $. We call a tuple $(\M, g)$ to be a Riemannian manifold.

In any smooth local coordinate $\{x^i\}$, a Riemannian metric can be written as tensor product

$$g = g_{ij} \, dx^i \otimes dx^j \, ,$$

such that

$$g(v, w) = g_{ij} \, dx^i \otimes dx^j(v, w) = g_{ij} \, dx^i(v) dx^j(w) = g_{ij} \, v^i w^j \, .$$

That is we can represent $ g $ as a symmetric, positive definite matrix $ G $ taking two tangent vectors as its arguments: $ \inner{v, w}_g = v^\text{T} G w $. Furthermore, we can define a norm w.r.t. $g$ to be $\norm{\cdot}_g := \inner{v, v}_g$ for any $v \in T_p \M$.

Example 8 (The Euclidean Metric). The simplest example of a Riemannian metric is the familiar Euclidean metric $g$ of $\R^n$ using the standard coordinate. It is defined by

$$g = \delta_{ij} \, dx^i \otimes dx^j \, ,$$

which, if applied to vectors $v, w \in T_p \R^n$, yields

$$g_p(v, w) = \delta_{ij} \, v^i w^j = \sum_{i=1}^n v^i w^i = v \cdot w \, .$$

Note that above, $\delta_{ij}$ is the Kronecker delta. Thus, the Euclidean metric can be represented by the $n \times n$ identity matrix.

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The tangent-cotangent isomorphism

Riemannian metrics also provide an isomorphism between the tangent and cotangent space: They allow us to convert vectors to covectors and vice versa. Let $(\M, g)$ be a Riemannian manifold. We define an isomorphism $\hat{g}: T_p \M \to T_p^* \M$ as follows. For each $p \in \M$ and each $v \in T_p \M$

$$\hat{g}(v) = \inner{v, \cdot}_g \, .$$

Notice that that $\hat{g}(v)$ is in $T_p^* \M$ as it is a linear functional over $T_p \M$. In any smooth coordinate $\{x^i\}$, by definition we can write $g = g_{ij} \, dx^i dx^j$. Thus we can write the isomorphism above as

$$\hat{g}(v) = (g_{ij} \, v^i) \, dx^j =: v_i \, dx^j \, .$$

Notice that we transform a contravariant component $v^i$ (denoted by the upper index component $i$) to a covariant component $v_i = g_{ij} \, v^i$ (denoted by the lower index component $j$), with the help of the metric tensor $g$. Because of this, we say that we obtain a covector from a tangent vector by lowering an index. Note that, we can also denote this by “flat” symbol in musical sheets: $\hat{g}(v) =: v^\flat$.

As Riemannian metric can be seen as a symmetric positive definite matrix, it has an inverse $g^{ij} := g_{ij}^{-1}$, which we denote by moving the index to the top, such that $g^{ij} \, g_{jk} = g_{kj} \, g^{ji} = \delta^i_k$. We can then define the inverse map of the above isomorphism as $\hat{g}^{-1}: T_p^* \M \to T_p \M$, where

$$\hat{g}^{-1}(\omega) = (g^{ij} \, \omega_j) \, \frac{\partial}{\partial x^i} =: \omega^i \, \frac{\partial}{\partial x^i} \, ,$$

for all $\omega \in T_p^* \M$. In correspondence with the previous operation, we are now looking at the components $\omega^i := g^{ij} \, \omega_j$, hence this operation is called raising an index, which we can also denote by “sharp” musical symbol: $\hat{g}^{-1}(\omega) =: \omega^\sharp$. Putting these two map together, we call the isomorphism between the tangent and cotangent space as the musical isomorphism.

The Riemannian gradient

Let $(\M, g)$ be a Riemannian manifold, and let $f: \M \to \R$ be a real-valued function over $\M$ (i.e. a scalar field on $\M)$. Recall that $df$ is a covector field, which in coordinates has partial derivatives of $f$ as its components. We define a vector field called the gradient of $f$ by

$$\begin{align} \grad{f} := (df)^\sharp = \hat{g}^{-1}(df) \, . \end{align}$$

For any $p \in \M$ and for any $v \in T_p \M$, the gradient satisfies

$$\inner{\grad{f}, v}_g = vf \, .$$

That is, for each $p \in \M$ and for any $v \in T_p \M$, $\grad{f}$ is a vector in $T_p \M$ such that the inner product with $v$ is the derivation of $f$ by $v$. Observe the compatibility of this definition with standard multi-variable calculus: the directional derivative of a function in the direction of a vector is the dot product of its gradient and that vector.

In any smooth coordinate $\{x^i\}$, $\grad{f}$ has the expression

$$\grad{f} = g^{ij} \frac{\partial f}{\partial x^i} \frac{\partial}{\partial x^j} \, .$$

Example 9 (Euclidean gradient). On $\R^n$ with the Euclidean metric with the standard coordinate, the gradient of $f: \R^n \to \R$ is

$$\grad{f} = \delta^{ij} \, \frac{\partial f}{\partial x^i} \frac{\partial}{\partial x^j} = \sum_{i=1}^n \frac{\partial f}{\partial x^i} \frac{\partial}{\partial x^i} \, .$$

Thus, again it is coincide with the definition we are familiar with form calculus.

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All in all then, given a basis, in matrix notation, let $G$ be the matrix representation of $g$ and let $d$ be the matrix representation of $df$ (i.e. as a row vector containing all partial derivatives of $f$), then: $\grad{f} = G^{-1} d^\T$.

The interpretation of the gradient in Riemannian manifold is analogous to the one in Euclidean space: its direction is the direction of steepest ascent of $f$ and it is orthogonal to the level sets of $f$; and its length is the maximum directional derivative of $f$ in any direction.

Connections

Let $(\M, g)$ be a Riemannian manifold and let $X, Y: \M \to T \M$ be a vector field. Applying the usual definition for directional derivative, the way we differentiate $X$ is by

$$D_X \vert_p Y = \lim_{h \to 0} \frac{Y_{p+hX_p} - Y_p}{h} \, .$$

However, we will have problems: We have not defined what this expression $p+hX_p$ means. Furthermore, as $Y_{p+hX_p}$ and $Y_p$ live in different vector spaces $T_{p+hX_p} \M$ and $T_p \M$, it does not make sense to subtract them, unless there is a natural isomorphism between each $T_p \M$ and $\M$ itself, as in Euclidean spaces. Hence, we need to add an additional structure, called connection that allows us to compare different tangent vectors from different tangent spaces of nearby points.

Specifically, we define the affine connection to be a connection in the tangent bundle of $\M$. Let $\mathfrak{X}(\M)$ be the space of vector fields on $\M$; $X, Y, Z \in \mathfrak{X}(\M)$; $f, g \in C^\infty(\M)$; and $a, b \in \R$. The affine connection is given by the map

$$\begin{align} \nabla: \mathfrak{X}(\M) \times \mathfrak{X}(\M) &\to \mathfrak{X}(\M) \\ (X, Y) &\mapsto \nabla_X Y \, , \end{align}$$

which satisfies the following properties

1. $C^\infty(\M)$-linearity in $X$, i.e., $\nabla_{fX+gY} Z = f \, \nabla_X Z + g \, \nabla_Y Z$
2. $\R$-linearity in Y, i.e., $\nabla_X (aY + bZ) = a \, \nabla_X Y + b \, \nabla_X Z$
3. Leibniz rule, i.e., $\nabla_X (fY) = (Xf) Y + f \, \nabla_X Y$ .

We call $\nabla_X Y$ the covariant derivative of $Y$ in the direction $X$. Note that the notation $Xf$ means $Xf(p) := D_{X_p} \vert_p f$ for all $p \in \M$, i.e. the directional derivative (it is a scalar field). Furthermore, notice that, covariant derivative and connection are the same thing and they are useful to generalize the notion of directional derivative to vector fields.

In any smooth local frame $(E_i)$ in $T \M$ on an open subset $U \in \M$, we can expand the vector field $\nabla_{E_i} E_j$ in terms of this frame

$$\nabla_{E_i} E_j = \Gamma^k_{ij} E_k \,.$$

The $n^3$ smooth functions $\Gamma^k_{ij}: U \to \R$ is called the connection coefficients or the Christoffel symbols of $\nabla$.

Example 10 (Covariant derivative in Euclidean spaces). Let $\R^n$ with the Euclidean metric be a Riemannian manifold. Then

$$(\nabla_Y X)_p = \lim_{h \to 0} \frac{Y_{p+hX_p} - Y_p}{h} \enspace \enspace \forall p \in \M \, ,$$

the usual directional derivative, is a covariant derivative.

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There exists a unique affine connection for every Riemannian manifold $(\M, g)$ that satisfies

1. Symmetry, i.e., $\nabla_X Y - \nabla_Y X = [X, Y]$
2. Metric compatible, i.e., $Z \inner{X, Y}_g = \inner{\nabla_Z X, Y}_g + \inner{X, \nabla_Z Y}_g$,

for all $X, Y, Z \in \mathfrak{X}(\M)$. It is called the Levi-Civita connection. Note that, $[\cdot, \cdot]$ is the Lie bracket, defined by $[X, Y]f = X(Yf) - Y(Xf)$ for all $f \in C^\infty(\M)$. Note also that, the connection shown in Example 10 is the Levi-Civita connection for Euclidean spaces with the Euclidean metric.

Riemannian Hessians

Let $(\M, g)$ be a Riemannian manifold equipped by the Levi-Civita connection $\nabla$. Given a scalar field $f: \M \to \R$ and any $X, Y \in \mathfrak{X}(\M)$, the Riemannian Hessian of $f$ is the covariant 2-tensor field $\text{Hess} \, f := \nabla^2 f := \nabla \nabla f$, defined by

$$\text{Hess} \, f(X, Y) := X(Yf) - (\nabla_X Y)f = \inner{\nabla_X \, \grad{f}, Y}_g \, .$$

Another way to define Riemannian Hessian is to treat is a linear map $T_p \M \to T_p \M$, defined by

$$\text{Hess}_{v} \, f = \nabla_v \, \grad{f} \, ,$$

for every $p \in \M$ and $v \in T_p \M$.

In any local coordinate $\{x^i\}$, it is defined by

$$\text{Hess} \, f = f_{; i,j} \, dx^i \otimes dx^j := \left( \frac{\partial f}{\partial x^i \partial x^j} - \Gamma^k_{ij} \frac{\partial f}{\partial x^k} \right) \, dx^i \otimes dx^j \, .$$

Example 11 (Euclidean Hessian). Let $\R^n$ be a Euclidean space with the Euclidean metric and standard Euclidean coordinate. We can show that connection coefficients of the Levi-Civita connection are all $0$. Thus the Hessian is defined by

$$\text{Hess} \, f = \left( \frac{\partial f}{\partial x^i \partial x^j} \right) \, dx^i \otimes dx^j \, .$$

It is the same Hessian as we have seen in calculus.

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Geodesics

Let $(\M, g)$ be a Riemannian manifold and let $\nabla$ be a connection on $T\M$. Given a smooth curve $\gamma: I \to \M$, a vector field along $\gamma$ is a smooth map $V: I \to T\M$ such that $V(t) \in T_{\gamma(t)}\M$ for every $t \in I$. We denote the space of all such vector fields $\mathfrak{X}(\gamma)$. A vector field $V$ along $\gamma$ is said to be extendible if there exists another vector field $\tilde{V}$ on a neighborhood of $\gamma(I)$ such that $V = \tilde{V} \circ \gamma$.

For each smooth curve $\gamma: I \to \M$, the connection determines a unique operator

$$D_t: \mathfrak{X}(\gamma) \to \mathfrak{X}(\gamma) \, ,$$

called the covariant derivative along $\gamma$, satisfying (i) linearity over $\R$, (ii) Leibniz rule, and (iii) if it $V \in \mathfrak{X}(\gamma)$ is extendible, then for all $\tilde{V}$ of $V$, we have that $ D_t V(t) = \nabla_{\gamma’(t)} \tilde{V}$.

For every smooth curve $\gamma: I \to \M$, we define the acceleration of $\gamma$ to be the vector field $D_t \gamma’$ along $\gamma$. A smooth curve $\gamma$ is called a geodesic with respect to $\nabla$ if its acceleration is zero, i.e. $D_t \gamma’ = 0 \enspace \forall t \in I$. In term of smooth coordinates $\{x^i\}$, if we write $\gamma$ in term of its components $\gamma(t) := \{x^1(t), \dots, x^n(t) \}$, then it follows that $\gamma$ is a geodesic if and only if its component functions satisfy the following geodesic equation:

$$\ddot{x}^k(t) + \dot{x}^i(t) \, \dot{x}^j(t) \, \Gamma^k_{ij}(x(t)) = 0 \, ,$$

where we use $x(t)$ as an abbreviation for $\{x^1(t), \dots, x^n(t)\}$. Observe that, this gives us a hint that to compute a geodesic we need to solve a system of second-order ODE for the real-valued functions $x^1, \dots, x^n$.

Suppose $\gamma: [a, b] \to \M$ is a smooth curve segment with domain in the interval $[a, b]$. The length of $\gamma$ is

$$L_g (\gamma) := \int_a^b \norm{\gamma'(t)}_g \, dt \, ,$$

where $\gamma’$ is the derivative (the velocity vector) of $\gamma$. We can then use curve segments as “measuring tapes” to measure the Riemannian distance from $p$ to $q$ for any $p, q \in \M$$

$$d_g(p, q) := \inf \, \{L_g(\gamma) \, \vert \, \gamma: [a, b] \to \M \enspace \text{s.t.} \enspace \gamma(a) = p, \, \gamma(b) = q\} \, ,$$

over all curve segments $\gamma$ which have endpoints at $p$ and $q$. We call the particular $\gamma$ such that $L_g(\gamma) = d_g(p, q)$ as the length-minimizing curve. We can show that all geodesics are locally length-minimizing, and all length-minimizing curves are geodesics.

Parallel transport

Let $(\M, g)$ be a Riemannian manifold with affine connection $\nabla$. A smooth vector field $V$ along a smooth curve $\gamma: I \to \M$ is said to be parallel along $\gamma$ if $D_t V = 0$ for all $t \in I$. Notice that a geodesic can then be said to be a curve whose velocity vector field is parallel along the curve.

Given $t_0 \in I$ and $v \in T_{\gamma(t_0)} \M$, we can show there exists a unique parallel vector field $V$ along $\gamma$ such that $V(t_0) = v$. This vector field is called the parallel transport of $v$ along $\gamma$. Now, for each $t_0, t_1 \in I$, we define a map

$$\begin{align} &P^\gamma_{t_0 t_1} : T_{\gamma(t_0)} \M \to T_{\gamma(t_1)} \M \\ &P^\gamma_{t_0 t_1}(v) = V(t_1) \, , \end{align}$$

called the parallel transport map. We can picture the concept of parallel transport by imagining that we are “sliding” a tangent vector $v$ along $\gamma$ such that the direction and the magnitude of $v$ is preserved.

Note that, the parallel transport map is a linear map with inverse $P^\gamma_{t_1 t_0}$, hence it is an isomorphism between two tangent spaces $T_{\gamma(t_0)} \M$ and $T_{\gamma(t_1)} \M$. We can therefore determine the covariant derivative along $\gamma$ using parallel transport:

$$D_t V(t_0) = \lim_{t_1 \to t_0} \frac{P^\gamma_{t_1 t_0} \, V(t_1) - V(t_0)}{t_1 - t_0} \, ,$$

Moreover, we can also determine the connection $\nabla$ via parallel transport:

$$\nabla_X Y \, \vert_p = \lim_{h \to 0} \frac{P^\gamma_{h 0} Y_{\gamma(h)} - Y_p}{h} \, ,$$

for every $p \in \M$.

Finally, if $A$ is a smooth vector field on $\M$, then $A$ is parallel on $\M$ if and only if $\nabla A = 0$.

The exponential map

Geodesics with proportional initial velocities are related in a simple way. Let $(\M, g)$ be a Riemannian manifold equipped with the Levi-Civita connection. For every $p \in \M$, $v \in T_p \M$, and $c, t \in \R$,

$$\gamma_{cv} (t) = \gamma_{v} (ct) \, ,$$

whenever either side is defined. This fact is compatible with our intuition on how speed and time are related to distance.

From the fact above, we can define a map from the tangent bundle to $\M$ itself, which sends each line through the origin in $T_p \M$ to a geodesic. Define a subset $\mathcal{E} \subseteq T\M$, the domain of the exponential map by

$$\mathcal{E} := \{ v \in T\M : \gamma_v \text{ is defined on an interval containing } [0, 1] \} \, ,$$

and then define the exponential map

$$\begin{align} &\text{exp}: \mathcal{E} \to \M \\ &\text{exp}(v) = \gamma_v(1) \, . \end{align}$$

For each $p \in \M$, the restricted exponential map at $p$, denoted $\text{exp}_p$ is the restriction of $\text{exp}$ to the set $\mathcal{E}_p := \mathcal{E} \cap T_p \M$.

The interpretation of the (restricted) exponential maps is that, given a point $p$ and tangent vector $v$, we follow a geodesic which has the property $\gamma(0) = p$ and $\gamma’(0) = v$. This is then can be seen as the generalization of moving around the Euclidean space by following straight line in the direction of velocity vector.

Curvature

Let $(\M, g)$ be a Riemannian manifold. Recall that an isometry is a map that preserves distance. Now, $\M$ is said to be flat if it is locally isometric to a Euclidean space, that is, every point in $\M$ has a neighborhood that is isometric to an open set in $\R^n$. We say that a connection $\nabla$ on $\M$ satisfies the flatness criterion if whenever $X, Y, Z$ are smooth vector fields defined on an open subset of $\M$, the following identity holds:

$$\nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z = \nabla_{[X, Y]} Z \, .$$

Furthermore, we can show that $(\M, g)$ is a flat Riemannian manifold, then its Levi-Civita connection satisfies the flatness criterion.

Example 12 (Euclidean space is flat). Let $\R^n$ with the Euclidean metric be a Riemannian manifold, equipped with the Euclidean connection $\nabla$. Then, given $X, Y, Z$ smooth vector fields:

$$\begin{align} \nabla_X \nabla_Y Z &= \nabla_X (Y(Z^k) \partial_k) = XY(Z^k) \partial_k \\ \nabla_Y \nabla_X Z &= \nabla_Y (X(Z^k) \partial_k) = YX(Z^k) \partial_k \, . \end{align}$$

The difference between them is

$$(XY(Z^k) - YX(Z^k)) \partial_k = \nabla_{[X, Y]}Z \, ,$$

by definition. Thus

$$\nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z = \nabla_{[X, Y]}Z \, .$$

Therefore, the Euclidean space with the Euclidean connection (which is the Levi-Civita connection on Euclidean space) is flat.

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Based on the above definition of the flatness criterion, then we can define a measure on how far away a manifold to be flat:

$$\begin{align} &R: \mathfrak{X}(\M) \times \mathfrak{X}(\M) \times \mathfrak{X}(\M) \to \mathfrak{X}(\M) \\ &R(X, Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X, Y]} Z \, , \end{align}$$

which is a multilinear map over $C^\infty (\M)$, and is therefore a $(1, 3)$-tensor field on $\M$.

We can then define a covariant 4-tensor called the (Riemann) curvature tensor to be the $(0, 4)$-tensor field $Rm := R^\flat$, by lowering the contravariant index of $R$. Its action on vector fields is given by

$$Rm(X, Y, Z, W) := \inner{R(X, Y)Z, W}_g \, .$$

In any local coordinates, it is written

$$Rm = R_{ijkl} \, dx^i \otimes dx^j \otimes dx^k \otimes dx^l \, ,$$

where $R_{ijkl} = g_{lm} \, {R_{ijkl}}^m$. We can show that $Rm$ is a local isometry invariant. Furthermore, compatible with our intuition of the role of the curvature tensor, a Riemannian manifold is flat if and only if its curvature tensor vanishes identically.

Working with $4$-tensors are complicated, thus we want to construct simpler tensors that summarize some of the information contained in the curvature tensor. For that, first we need to define the trace operator for tensors. Let $T^{(k,l)}(V)$ denotes the space of tensors with $k$ covariant and $l$ contravariant components of a vector space $V$, the trace operator is:

$$\begin{align} &\text{tr}: T^{(k+1, l+1)}(V) \to T^{(k,l)}(V) \\ &(\text{tr} \, F)(\omega^1, \dots, \omega^k, v_1, \dots, v_l) = \text{tr}(F(\omega^1, \dots, \omega^k, \cdot, v_1, \dots, v_l, \cdot)) \, , \end{align}$$

where the trace operator in the right hand side is the usual trace operator, as $F(\omega^1, \dots, \omega^k, \cdot, v_1, \dots, v_l, \cdot) \in T^{(1,1)}(V)$ is a $(1,1)$-tensor, which can be represented by a matrix. We can extend this operator to covariant tensors in Riemannian manifolds: If $h$ is any covariant $k$-tensor field with $k \geq 2$, we can raise one of its indices and obtain $(1, k-1)$-tensor $h^\sharp$. The trace of $h^\sharp$ is thus a covariant $(k-2)$-tensor field. All in all, we define the trace of $h$ w.r.t. $g$ as

$$\text{tr}_g \, h := \text{tr}(h^\sharp) \, .$$

In coordinates, it is

$$\text{tr}_g \, h = {h_i}^i = g^{ij} h_{ij} \, ,$$

which, in an orthonormal frame, it is given by the ordinary trace of the matrix $(h_{ij})$.

We now define the Ricci curvature or Ricci tensor $Rc$ which is the covariant 2-tensor field defined as follows:

$$Rc(X, Y) := \text{tr}(Z \mapsto R(Z, X)Y) \, ,$$

for any vector fields $X, Y$. In local coordinates, its components are

$$R_{ij} := {R_{kij}}^k = g^{km} \, R_{kijm} \, .$$

We can simplify it further: We define the scalar curvature to be the function $S$ to be the trace of the Ricci tensor:

$$S := \text{tr}_g \, Rc = {R_i}^i = g^{ij} \, R_{ij} \, .$$

Thus the scalar curvature is a scalar field on $\M$.

Submanifolds

Let $\M$ be a smooth manifold. An embedded or regular submanifold of $\M$ is a subset $\mathcal{S} \subset \M$ that is a manifold in the subspace topology, endowed with a smooth structure w.r.t. which the inclusion map $\mathcal{S} \hookrightarrow \M$ is a smooth embedding. We call the difference $\text{dim} \, \M - \text{dim} \, \mathcal{S}$ to be the codimension of $\mathcal{S}$ in $\M$, and $\M$ to be the ambient manifold. An embedded hypersurface is an embedded submanifold of codimension 1.

Example 13 (Graphs as submanifolds). Suppose $\M$ is a smooth $m$-manifold, $\mathcal{N}$ is a smooth $n$-manifold, $U \subset \M$ is open, and $f: U \to \mathcal{N}$ is a smooth map. Let $\Gamma(f) \subseteq \M \times \mathcal{N}$ denote the graph of $f$, i.e.

$$\Gamma(f) := \{ (x, y) \in \M \times \mathcal{N} : x \in U, y = f(x) \} \, .$$

Then $\Gamma(f)$ is an embedded $m$-submanifold of $\M \times \mathcal{N}$.

Furthermore, if $f: \M \to \mathcal{N}$ is a smooth map (notice that we have defined $f$ globally here), then $\Gamma(f)$ is properly embedded in $\M \times \mathcal{N}$, i.e. the inclusion map is a proper map.

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Suppose $\M$ and $\N$ are smooth manifolds. Let $F: \M \to \N$ be a smooth map and $p \in \M$. We define the rank of $F$ at $p$ to be the rank of the linear map $dF_p: T_p\M \to T_{F(p)\N}$, i.e. the rank of the Jacobian matrix of $F$ in coordinates. If $F$ has the same rank $r$ at any point, we say that it has constant rank, written $\rank{F} = r$. Note that it is bounded by $\min \{ \dim{\M}, \dim{\N} \}$ and if it is equal to this bound, we say $F$ has full rank at $p$.

A smooth map $F: \M \to \N$ is called a smooth submersion if $dF$ is surjective at each point ($\rank{F} = \dim{\N}$). It is called a smooth immersion if $dF$ is injective at each point ($\rank{F} = \dim{\M}$).

Example 14 (Submersions and immersions).

1. Suppose $\M_1, \dots, \M_k$ are smooth manifolds. Then each projection maps $\pi_i: \M_1 \times \dots \times \M_k \to \M_i$ is a smooth submersion. In particular $\pi: \R^{n+k} \to \R^n$ is a smooth submersion.
2. If $\gamma: I \to \M$ is a smooth curve in a smooth manifold $\M$, then $\gamma$ is a smooth immersion if and only if $\gamma’(t) \neq 0$ for all $t \in I$.

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If $\M$ and $\N$ are smooth manifolds. A diffeomorphism from $\M$ to $\N$ is a smooth bijective map $F: \M \to \N$ that has a smooth inverse, and $\M$ and $\N$ are said to be diffeomorphic. $F$ is called a local diffeomorphism if every point $p \in \M$ has a neighborhood $U$ such that $F(U)$ is open in $\N$ and $F\vert_U: U \to F(U)$ is a diffeomorphism. We can show that $F$ is a local diffeomorphism if and only if it is both a smooth immersion and submersion. Furthermore, if $\dim{\M} = \dim{\N}$ and $F$ is either a smooth immersion or submersion, then it is a local diffeomorphism.

The Global rank theorem says that if $\M$ and $\N$ are smooth manifolds, and suppose $F: \M \to \N$ is a smooth map of constant rank, then it is (a) a smooth submersion if it is injective, (b) a smooth immersion if it is injective, and (c) a diffeomorphism if it is bijective.

If $\M$ and $\N$ are smooth manifolds, a smooth embedding of $\M$ into $\N$ is a smooth immersion $F: \M \to \N$ that is also a topological embedding (homeomorphism onto its image in the subspace topology).

Example 15 (Smooth embeddings). If $\M$ is a smooth manifold and $U \subseteq \M$ is an open submanifold, the inclusion $U \hookrightarrow \M$ is a smooth embedding.

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Let $F: \M \to \N$ be an injective smooth immersion. If any of these condition holds, then $F$ is a smooth embedding: (a) $F$ is an open or closed map, (b) $F$ is a proper map, (c) $\M$ is compact, and (d) $\M$ has empty boundary and $\dim{\M} = \dim{\N}$.

The second fundamental form

Let $(\M, g)$ be a Riemannian submanifold of a Riemannian manifold $(\tilde{\M}, \tilde{g})$. Then, $g$ is the induced metric $g = \iota_\M^* \tilde{g}$, where $\iota_\M: \M \hookrightarrow \tilde{\M}$ is the inclusion map. Note that, the expression $\iota^*_\M \tilde{g}$ is called the pullback metric or the induced metric of $\tilde{g}$ by $\iota_\M$ and is defined by

$$\iota_\M^* \tilde{g}(u, v) := \tilde{g}(d\iota_\M(u), d\iota_\M(v)) \, ,$$

for any $u, v \in T_p \M$. Also, recall that $d\iota_\M$ is the pushforward (tangent map) by $\iota_\M$. Intuitively, we map the tangent vectors $u, v$ of $T_p \M$ to some tangent vectors of $T_{\iota_\M(p)} \tilde{\M}$ and use $\tilde{g}$ as the metric.

In this section, we will denote any geometric object of the ambient manifold with tilde, e.g. $\tilde{\nabla}, \tilde{Rm}$, etc. Note also that, we can use the inner product notation $\inner{u, v}$ to refer to $g$ or $\tilde{g}$, since $g$ is just the restriction of $\tilde{g}$ to pairs of tangent vectors in $T \M$.

We would like to compare the Levi-Civita connection of $\M$ with that of $\tilde{\M}$. First, we define orthogonal projection maps, called tangential and normal projections by

$$\begin{align} \pi^\top &: T \tilde{\M} \vert_\M \to T\M \\ \pi^\perp &: T \tilde{\M} \vert_\M \to N\M \, , \end{align}$$

where $N\M$ is the normal bundle of $\M$, i.e. the set of all vectors normal to $\M$. If $X$ is a section of $T\tilde{\M}\vert_\M$, we use the shorthand notations $X^\top = \pi^\top X$ and $X^\perp = \pi^\perp X$.

Given $X, Y \in \mathfrak{X}(\M)$, we can extend them to vector fields on an open subset of $\tilde{\M}$, apply the covariant derivative $\tilde{\nabla}$, and then decompose at $p \in \M$ to get

$$\tilde{\nabla}_X Y = (\tilde{\nabla}_X Y)^\top + (\tilde{\nabla}_X Y)^\perp \, .$$

Let $\Gamma(E)$ be the space of smooth sections of bundle $E$. For the second part, we define the second fundamental form of $\M$ to be a map $\two: \mathfrak{X}(\M) \times \mathfrak{X}(\M) \to \Gamma(N\M)$ defined by

$$\two(X, Y) = (\tilde{\nabla}_X Y)^\perp \, .$$

Meanwhile, we can show that, the first part is the covariant derivative w.r.t. the Levi-Civita connection of the induced metric on $\M$. All in all, the above equation can be written as the Gauss formula:

$$\tilde{\nabla}_X Y = \nabla_X Y + \two(X, Y) \, .$$

The second fundamental form can also be used to evaluate extrinsic covariant derivatives of normal vector fields (instead of tangent ones above). For each normal vector field $N \in \Gamma(N\M)$, we define a scalar-valued symmetric bilinear form $\two_N: \mathfrak{X}(\M) \times \mathfrak{X}(\M) \to \R$ by

$$\two_N(X, Y) = \inner{N, \two(X, Y)} \, .$$

Let $W_N: \mathfrak{X}(\M) \to \mathfrak{X}(\M)$ denote the self-adjoint linear map associated with this bilinear form, characterized by

$$\inner{W_N(X), Y} = \two_N(X, Y) = \inner{N, \two(X, Y)} \, .$$

The map $W_N$ is called the Weingarten map in the direction of $N$. Furthermore we can show that the equation $(\tilde{\nabla}_X N)^\top = -W_N(X)$ holds and is called the Weingarten equation.

In addition to describing the difference between the intrinsic and extrinsic connections, the second fundamental form describes the difference between the curvature tensors of $\tilde{\M}$ and $\M$. The explicit formula is called the Gauss equation and is given by

$$\tilde{Rm}(W, X, Y, Z) = Rm(W, X, Y, Z) - \inner{\two(W, Z), \two(X, Y)} + \inner{\two(W, Y), \two(X, Z)} \, .$$

To give a geometric interpretation of the second fundamental form, we study the curvatures of curves. Let $\gamma: I \to \M$ be a smooth unit-speed curve. We define the curvature of $\gamma$ as the length of the acceleration vector field, i.e. the function $\kappa: I \to \R$ given by $\kappa(t) := \norm{D_t \gamma’(t)}$. We can see this curvature of the curve as a quantitative measure of how far the curve deviates from being a geodesic. Note that, if $\M = \R^n$ the curvature agrees with the one defined in calculus.

Now, suppose that $\M$ is a submanifold in the ambient manifold $\tilde{\M}$. Every regular curve $\gamma: I \to \M$ has two distinct curvature: its intrinsic curvature $\kappa$ as a curve in $\M$ and its extrinsic curvature $\tilde{\kappa}$ as a curve in $\tilde{\M}$. The second fundamental form can then be used to compute the relationship between the two: For $p \in \M$ and $v \in T_p \M$, (i) $\two(v, v)$ is the $\tilde{g}$-acceleration at $p$ of the $g$-geodesic $\gamma_v$, and (ii) if $v$ is a unit vector, then $\norm{\two(v, v)}$ is the $\tilde{g}$-curvature of $\gamma_v$ at $p$.

The intrinsic and extrinsic accelerations of a curve are usually different. A Riemannian submanifold $(\M, g)$ of $(\tilde{\M}, \tilde{g})$ is said to be totally geodesic if every $\tilde{g}$-geodesic that is tangent to $\M$ at some time $t_0$ stays in $\M$ for all $t \in (t_0 - \epsilon, t_0 + \epsilon)$.

Riemannian hypersurfaces

We focus on the case when $(\M, g)$ is an embedded $n$-dimensional Riemannian submanifold of an $(n+1)$-dimensional Riemannian manifold $(\tilde{\M}, \tilde{g})$. That is, $\M$ is a hypersurface of $\tilde{\M}$.

In this situation, at each point of $\M$, there are exactly two unit normal vectors. We choose one of these normal vector fields and call it $N$. We can replace the vector-valued second fundamental form above by a simpler scalar-valued form. The scalar second fundamental form of $\M$ is the symmetric covariant $2$-tensor field $h = \two_N$, i.e.

$$h(X, Y) := \inner{N, \two(X, Y)} \enspace \enspace \enspace \text{for all } X, Y \in \mathfrak{X}(\M) \, .$$

By the Gauss formula $\tilde{\nabla}_X Y = \nabla_X Y + \two(X, Y)$ and noting that $\nabla_X Y$ is orthogonal to $N$, we can rewrite the definition as $h(X, Y) = \inner{N, \tilde{\nabla}_X Y}$. Furthermore, since $N$ is a unit vector spanning $N\M$, we can write $\two(X, Y) = h(X, Y)N$. Note that the sign of $h$ depends on the normal vector field chosen.

The choice of $N$ also determines a Weingarten map $W_N: \mathfrak{X}(\M) \to \mathfrak{X}(\M)$. In this special case of a hypersurface, we use the notation $s = W_N$ and call it the shape operator of $\M$. We can think of $s$ as the $(1, 1)$-tensor field on $\M$ obtained from $h$ by raising an index. It is characterized by

$$\inner{sX, Y} = h(X, Y) \enspace \enspace \enspace \text{for all } X, Y \in \mathfrak{X}(\M) \, .$$

As with $h$, the choice of $N$ determines the sign of $s$.

Note that at every $p \in \M$, $s$ is a self-adjoint linear endomorphism of the tangent space $T_p \M$. Let $v \in T_p \M$. From linear algebra, we know that there is a unit vector $v_0 \in T_p \M$ such that $v \mapsto \inner{sv, v}$ achieve its maximum among all unit vectors. Every such vector is an eigenvector of $s$ with eigenvalue $\lambda_0 = \inner{s v_0, v_0}$. Furthermore, $T_p \M$ has an orthonormal basis $(b_1, \dots, b_n)$ formed by the eigenvectors of $s$ and all of the eigenvalues $(\kappa_1, \dots \kappa_n)$ are real. (Note that this means for each $i$, $s b_i = \kappa_i b_i)$.) In this basis, both $h$ and $s$ are represented by diagonal matrices.

The eigenvalues of $s$ at $p \in \M$ are called the principal curvatures of $\M$ at $p$, and the corresponding eigenvectors are called the principal directions. Note that the sign of the principal curvatures depend on the choice of $N$. But otherwise both the principal curvatures and directions are independent of the choice of coordinates.

From the principal curvatures, we can compute other quantities: The Gaussian curvature which is defined as $K := \text{det}(s)$ and the mean curvature $H := (1/n) \text{tr}(s)$. In other words, $K = \prod_i \kappa_i$ and $H = (1/n) \sum_i \kappa_i$, since $s$ can be represented by a symmetric matrix.

The Gaussian curvature, which is a local isometric invariant, is connected to a global topological invariant, the Euler characteristic, through the Gauss-Bonnet theorem. Let $(\M, g)$ be a smoothly triangulated compact Riemannian 2-manifold, then

$$\int_\M K \, dA = 2 \pi \, \chi(\M) \, ,$$

where $dA$ is its Riemannian density.

Hypersurfaces of Euclidean space

Assume that $\M \subseteq \R^{n+1}$ is an embedded Riemannian $n$-submanifold (with the induced metric from the Euclidean metric). We denote geometric objects on $\R^{n+1}$ with bar, e.g. $\bar{g}$, $\overline{Rm}$, etc. Observe that $\overline{Rm} \equiv 0$, which implies that the Riemann curvature tensor of a hypersurface in $\R^{n+1}$ is completely determined by the second fundamental form.

In this setting we can give some very concrete geometric interpretation about quantities in hypersurfaces. First is for curves. For every $v \in T_p \M$, let $\gamma = \gamma_v : I \to \M$ be the $g$-geodesic in $\M$ with initial velocity $v$. The Gauss formula shows that the Euclidean acceleration of $\gamma$ at $0$ is $\gamma^{\prime\prime}(0) = \overline{D}_t \gamma’(0) = h(v, v)N_p$, thus $\norm{h(v, v)}$ is the Euclidean curvature of $\gamma$ at $0$. Furthermore, $h(v,v) = \inner{\gamma^{\prime\prime}(0), N_p} > 0$ iff. $\gamma^{\prime\prime}(0)$ points in the same direction as $N_p$. That is $h(v, v)$ is positive if $\gamma$ is curving in the direction of $N_p$ and negative if it is curving away from $N_p$.

We can show that the above Euclidean curvature can be interpreted in terms f the radius of the “best circular approximation”, just in Calculus. Suppose $\gamma: I \to \R^m$ is a unit-speed curve, $t_0 \in I$, and $\kappa(t_0) \neq 0$. We define a unique unit-speed parametrized circle $c: \R \to \R^m$ as the osculating circle at $\gamma(t_0)$, with the property that $c$ and $\gamma$ have the same position, velocity, and acceleration at $t=t_0$. Then, the Euclidean curvature of $\gamma$ at $t_0$ is $\kappa(t_0) = 1/R$ where $R$ is the radius of the osculating circle.

As mentioned before, to compute the curvature of a hypersurface in Euclidean space, we can compute the second fundamental form. Suppose $X: U \to \M$ is a smooth local parametrization of $\M$, $(X_1, \dots, X_n)$ is the local frame for $T \M$ determined by $X$, and $N$ is a unit normal field on $\M$. Then, the scalar second fundamental form is given by

$$h(X_i, X_j) = \innerbig{\frac{\partial^2 X}{\partial u^i \partial u^j}, N} \, .$$

The implication of this is that it shows how the principal curvatures give a concise description of the local shape of the hypersurface by approximating the surface with the graph of a quadratic function. That is, we can show that there is an isometry $\phi: \R^{n+1} \to \R^{n+1}$ that takes $p \in \M$ to the origin and takes a neighborhood of it to a graph of the form $x^{n+1} = f(x^1, \dots, x^n)$, where

$$f(x) = \frac{1}{2} \sum_{i=1}^n\kappa_i (x^i)^2 + O(\abs{x}^3) \, .$$

We can write down a smooth vector field $N = N^i \partial_i$ on an open subset of $\R^{n+1}$ that restricts to a unit normal vector field along $\M$. Then, the shape operator can be computed straightforwardly using the Weingarten equation and observing that the Euclidean covariant derivatives of $N$ are just ordinary directional derivatives in Euclidean space. Thus, for every vector $X = X^i \partial_j$ tangent to $\M$, we have

$$sX = -\bar{\nabla}_X N = -\sum_{i,j=1}^{n+1} X^j (\partial_j N^i) \partial_i \, .$$

One common way to get such smooth vector field is to work with a local defining function $F$ for $\M$, i.e. a smooth scalar field defined on some open subset $U \subseteq \R^{n+1}$ s.t. $U \cap \M$ is a regular level set of $F$. Then, we can take

$$N = \frac{\grad{F}}{\norm{\grad{F}}} \, .$$

Because we know that the gradient is always normal to the level set.

Example 16 (Shape operators of spheres). The function $F: \R^{n+1} \to \R$ with $F(x) := \norm{x}^2$ is a smooth defining function of any sphere in $\mathbb{S}^{n}(R)$. Thus, the normalized gradient vector field

$$N = \frac{1}{R} \sum_{i,j=1}^{n+1} x^i \partial_i$$

is a (outward pointing) unit normal vector field along $\mathbb{S}^n(R)$. The shape operator is

$$sX = -\frac{1}{R} \sum_{i,j=1}^{n+1} X^j (\partial_j x^i) \partial_i = -\frac{1}{R} X \, ,$$

where recall that, $\partial_j x^i = \partial x^i / \partial x^j = \delta_{ij}$. We can therefore write $s$ as a matrix $s = (-1/R) \mathbf{I}$ where $\mathbf{I}$ is the identity matrix. The principal curvatures are then all equal to $-1/R$, the mean curvature is $H = -1/R$, and the Gaussian curvature is $K = (-1/R)^n$. Note that, these curvatures are constant. These reflects the fact that the sphere bends the exact same way at every point.

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Lastly, for surfaces in $\R^3$, given a parametrization of $X$, the normal vector field can be computed via the cross product:

$$N = \frac{X_1 \times X_2}{\norm{X_1 \times X_2}} \, ,$$

where $X_1 := \partial_1 X$ and $X_2 := \partial_2 X$, which together form a basis of the tangent space at each point on the surface.

Although the Gaussian curvature is defined in terms of a particular embedding of a submanifold in the Euclidean space (i.e. it is an extrinsic quantity), it is actually an intrinsic invariant of the submanifold. Gauss showed in his Theorema Egregium that in an embedded $2$-dimensional Riemannian submanifold $(\M, g)$ of $\R^3$, for every point $p \in \M$, the Gaussian curvature of $\M$ at $p$ is equal to one-half the scalar curvature of $g$ at $p$, and thus it is a local isometry invariant of $(\M, g)$.

Suppose $\M$ is a Riemannian $n$-manifold with $n \geq 2$, $p \in \M$, and $V \subset T_p \M$ is a star-shaped neighborhood of zero on which $\text{exp}_p$ is a diffeomorphism onto an open set $U \subset \M$. Let $\Pi$ be any $2$-dimensional linear subspace of $T_p \M$. Since $\Pi \cap V$ is an embedded $2$-dim submanifold of $V$, it follows that $\mathcal{S}_\Pi = \text{exp}_p(\Pi \cup V)$ is an embedded $2$-dim submmanifold of $U \subset \M$ containing $p$, called the plane section determined by $\Pi$. We define the sectional curvature of $\Pi$, denoted by $\text{sec}(\Pi)$, to be the intrinsic Gaussian curvature at $p$ of the surface $\mathcal{S}_\Pi$ with the metric induced from the embedding $\mathcal{S}_\Pi \subseteq \M$. If $v, w \in T_p \M$ are linearly independent vectors, the sectional curvature’s formula is given by

$$\text{sec}(v, w) := \frac{Rm_p(v, w, w, v)}{\norm{v \wedge w}^2} \, ,$$

where

$$\norm{v \wedge w} := \sqrt{\norm{v}^2 \norm{w}^2 - \inner{v, w}^2} \, .$$

We can show the connection between the sectional curvature and Ricci and scalar curvatures. $Rc_p(v, v)$ is the sum of the sectional curvatures of the $2$-planes spanned by $(v, b_2), \dots, (v, b_n)$, where $(b_1, \dots, b_n)$ is any orthonormal basis for $T_p \M$ with $b_1 = v$. Furthermore, the scalar curvature at $p$ is the sum of all sectional curvatures of the $2$-planes spanned by ordered pairs of distinct basis vectors in any orthonormal basis.

Lie groups

A Lie group is a smooth manifold $\G$ that is also a group in the algebraic sense, with the property that the multiplication map $m: \G \times \G \to \G$ and inversion map $i: \G \to \G$, given by

$$m(g, h) := gh \, , \qquad i(g) := g^{-1} \, ,$$

are both smooth for arbitrary $g, h \in \G$. We denote the identity element of $G$ by $e$.

Example 17 (Lie groups). The following manifolds are Lie groups.

1. The general linear group $\GL(n, \R)$ is the set of invertible $n \times n$ matrices with real elements. It is a group under matrix multiplication and it is a submanifold of the vector space $\text{M}(n, \R)$, the space of $n \times n$ matrices.

2. The real number field $\R$ and the Euclidean space $\R^n$ are Lie groups under addition.

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If $\G$ and $\mathcal{H}$ are Lie groups, a Lie group homomorphism from $\G$ to $\mathcal{H}$ is a smooth map $F: \G \to \mathcal{H}$ that is also a group homomorphism. If $F$ is also a diffeomorphism, then it is a Lie group isomorphism. We say that $\G$ and $\mathcal{H}$ are isomorphic Lie groups.

If $G$ is a group and $M$ is a set, a left action of $G$ on $M$ is a map $G \times M \to M$ defined by $(g, p) \mapsto g \cdot p$ that satisfies

$$\begin{alignat}{2} g_1 \cdot (g_2 \cdot p) &= (g_1 g_2) \cdot p \qquad &&\text{for all } g_1, g_2 \in G, p \in M \, ; \\ e \cdot p &= p &&\text{for all } p \in M \, . \end{alignat}$$

Analogously, a right action is defined as a map $M \times G \to M$ satisfying

$$\begin{alignat}{2} (p \cdot g_1) \cdot g_2 &= p \cdot (g_1 g_2) \qquad &&\text{for all } g_1, g_2 \in G, p \in M \, ; \\ p \cdot e &= p &&\text{for all } p \in M \, . \end{alignat}$$

If $M$ is a smooth manifold, $G$ is a Lie group, and the defining map is smooth, then the action is said to be smooth action.

We can also give a name to an action, e.g. $\theta: G \times M \to M$ with $(g, p) \mapsto \theta_g (p)$. In this notation, the above conditions for the left action read

$$\begin{align} \theta_{g_1} \circ \theta_{g_2} &= \theta_{g_1 g_2} \, , \\ \theta_e &= \Id_M \, , \end{align}$$

while for a right action the first equation is replaced by $\theta_{g_2} \circ \theta_{g_1} = \theta_{g_1 g_2}$. For a smooth action, each map $\theta_g : M \to M$ is a diffeomorphism.

For each $p \in M$, the orbit of $p$, denoted by $G \cdot p$, is the set of all images of $p$ under the action by elements of $G$:

$$G \cdot p := \{ g \cdot p : g \in G \} \, .$$

The isotropy group or stabilizer of $p$, denoted by $G_p$, is the set of elements of $G$ that fix $p$ (implying $G_p$ is a subgroup of $G$):

$$G_p := \{ g \in G : g \cdot p = p \} \, .$$

A group action is said to be transitive if for every pair of points $p, q \in M$, there exists $g \in G$ such that $g \cdot p = q$, i.e. if the only orbit is all of $M$. The action is said to be free if the only element of $G$ that fixes any element of $M$ is the identity: $g \cdot p$ for some $p \in M$ implies $g = e$, i.e. if every isotropy group is trivial.

Example 18 (Lie group actions).

1. If $\G$ is a Lie group and $\M$ is a smooth manifold, the trivial action of $\G$ on $\M$ is defined by $g \cdot p = p$ for all $g \in \G$ and $p \in \M$.

2. The natural action of $\GL(n, \R)$ on $\R^n$ is the left action given by matrix multiplication $(\b{A}, \vx) \mapsto \b{A} \vx$.

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Let $\G$ be a Lie group, $\M$ and $\N$ be smooth manifolds endowed with smooth left or right $\G$-actions. A map $F: \M \to \N$ is equivariant w.r.t. the given actions if for each $g \in G$,

$$\begin{alignat}{2} F(g \cdot p) &= g \cdot F(p) \qquad &&\text{for left actions} \, , \\ F(p \cdot g) &= F(p) \cdot g &&\text{for right actions} \, . \end{alignat}$$

If $F: \M \to \N$ is a smooth map that is equivariant w.r.t. a transitive smooth $\G$-action on $\M$ and any smooth $\G$-action on $\N$, then $F$ has constant rank, meaning that its rank is the same for all $p \in \M$. Thus, if $F$ is surjective, it is a smooth submersion; if it is injective, it is a smooth immersion; and if it is bijective, it is a diffeomorphism.

Example 19 (The orthogonal group). A real $n \times n$ matrix $\b{A}$ is said to be orthogonal if it preserves the Euclidean dot product as a linear map:

$$(\b{A} \vx) \cdot (\b{A} \vx) = \vx \cdot \vy \qquad \text{for all} \, \vx, \vy \in \R^n \, .$$

The set of all orthogonal $n \times n$ matrices $\text{O}(n)$ is a subgroup of $\GL(n, \R)$, called the orthogonal group of degree $n$.

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We would like to also study the theory of group representations, i.e. asking the question whether all Lie groups can be realized as Lie subgroups of $\GL(n, \R)$ or $\GL(n, \C)$. If $\G$ is a Lie group, a representation of $\G$ is a Lie group homomorphism from $\G$ to $\GL(V)$ for some finite-dimensional vector space $V$. Note that, $\GL(V)$ denotes the group of invertible linear transformations of $V$ which is a Lie group isomorphic to $\GL(n, \R)$. If a representation is injective, it is said to be faithful.

There is a close connection between representations and group actions. An action of $\G$ on $V$ is said to be a linear action if for each $g \in \G$, the map $V \to V$ defined by $x \mapsto g \cdot x$ is linear.

Example 20 (Linear action). If $\rho: \G \to \GL(V)$ is a representation of $\G$, there is an associated smooth linear action of $\G$ on $V$ given by $g \cdot x = \rho(g) x$. In fact, this holds for every linear action.

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References

1. Lee, John M. “Smooth manifolds.” Introduction to Smooth Manifolds. Springer, New York, NY, 2013. 1-31.
2. Lee, John M. Riemannian manifolds: an introduction to curvature. Vol. 176. Springer Science & Business Media, 2006.
3. Fels, Mark Eric. “An Introduction to Differential Geometry through Computation.” (2016).
4. Absil, P-A., Robert Mahony, and Rodolphe Sepulchre. Optimization algorithms on matrix manifolds. Princeton University Press, 2009.
5. Boumal, Nicolas. Optimization and estimation on manifolds. Diss. Catholic University of Louvain, Louvain-la-Neuve, Belgium, 2014.
6. Graphics: https://tex.stackexchange.com/questions/261408/sphere-tangent-to-plane.