## Tensors in Differential Geometry

Clayton Shonkwiler

shonkwiler.org

TACA 2019

/taca2019

This talk!

Funding: Simons Foundation

Colin Roberts

Harrison Chapman

### The Basic Problem

$$M$$

$$T_x M$$

$$x$$

$$y$$

$$u$$

$$v$$

$$T_y M$$

$$u$$ and $$v$$ live in different vector spaces!

### Vector Fields

The tangent bundle

\displaystyle TM := \bigsqcup_{x \in M} T_x M
M

A vector field on $$M$$ is a (smooth) section of the tangent bundle; i.e., a smooth map $$X: M \to TM$$ so that $$\pi \circ X= \operatorname{id}$$.

$$\pi$$

1. The collection $$\mathfrak{X}(M)$$ of all vector fields on $$M$$ forms a Lie algebra.
2. $$\mathfrak{X}(M)$$ is a $$C^\infty(M)$$-module.

Notes:

### Vector Fields as Derivations

$$M$$

$$x=\gamma(0)$$

$$\gamma'(0)=X(x)$$

$$\gamma(t)$$

$$X \in \mathfrak{X}(M)$$ is a differential operator: for $$f \in C^\infty(M)$$, $$X(f) \in C^\infty(M)$$ is defined by

X(f)(x) = (f \circ \gamma)'(0)

$$X(x)$$

$$x$$

The Lie bracket $$[X,Y](f) := X(Y(f))-Y(X(f))$$

### 1-Forms

The cotangent bundle

\displaystyle T^*M := \bigsqcup_{x \in M} (T_x M)^*
M

A $$1$$-form on $$M$$ is a (smooth) section of the cotangent bundle; i.e., a smooth map $$\omega: M \to T^*M$$ so that $$\pi \circ \omega = \operatorname{id}$$.

$$\pi$$

Example. If $$f: M \to \mathbb{R}$$ is a smooth function, then the differential $$df$$ is a 1-form.

The tensor space $$T^p_q(V)$$ is the collection of multilinear maps $$\underbrace{V^* \times \dots \times V^*}_p \times \underbrace{V \times \dots \times V}_q \rightarrowtail \mathbb{R}$$.

### Tensor Bundles

The tensor bundle

\displaystyle \mathcal{T}^p_q(M):= \sqcup_{x \in M} T^p_q(T_x M)

$$\pi$$

M
1.  $$\mathcal{T}^0_0(M)=C^\infty{M}$$
2. $$\mathcal{T}^1_0(M)=TM$$
3. $$\mathcal{T}^0_1(M) = T^*M$$

Implicitly using $$V^{**}\simeq V$$: think of $$v \in V$$ as a linear functional $$V^* \to \mathbb{R}$$ by $$v(\varphi):=\varphi(v)$$.

Examples.

(Can also think of $$\mathcal{T}^p_q(M)$$ as a tensor product of $$TM$$’s and $$T^*M$$’s over $$C^\infty(M)$$.)

### Tensor Fields

The tensor bundle

\displaystyle \mathcal{T}^p_q(M):= \bigsqcup_{x \in M} T^p_q(T_x M)

$$\pi$$

M

A $$(p,q)$$-tensor field (or just $$(p,q)$$-tensor) on $$M$$ is a (smooth) section of $$\mathcal{T}^p_q(M)$$.

Examples. Functions, vector fields, 1-forms.

Example. A Riemannian metric (or metric tensor) is a positive-definite symmetric $$(0,2)$$-tensor field $$g$$; i.e., $$g(x)=:g_x$$ is an inner product on $$T_xM$$ for each $$x$$.

### Riemannian Metrics

A Riemannian metric $$g$$ on $$M$$ is often written as

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

or just $$g_{ij}$$. The notation $$ds^2 = \sum g_{ij}dx^i dx^j = g_{ij}dx^idx^j$$ is also common.

Example. $$ds^2=dx^2 + dy^2$$ on $$\mathbb{R}^2$$ is just the standard dot product:

g\left(a \frac{\partial}{\partial x} + b \frac{\partial }{\partial y}, c \frac{\partial}{\partial x} + d \frac{\partial}{\partial y}\right) = ac+bd.

Example. $$ds^2 = \frac{1}{y^2}\left(dx^2 + dy^2\right)$$ on $$\mathbb{H}^2:=\{(x,y) \in \mathbb{R}^2 | y>0\}$$ gives the upper half-plane model of the hyperbolic plane.

### Riemannian Metrics and Lengths

Definition. Given a smooth curve $$\gamma: [a,b] \to M$$, where $$(M,g)$$ is a Riemannian manifold, then length of $$\gamma$$ is

\operatorname{Length}(\gamma) = \int_a^b \sqrt{g_{\gamma(t)}(\gamma'(t),\gamma'(t))} dt.

For $$x,y\in M$$, the distance from $$x$$ to $$y$$ is

d_g(x,y) := \inf \{\operatorname{Length}(\gamma)| \gamma: [0,1]\to M, \gamma(0)=x, \gamma(1)=y\}

### Differential Forms

A differential $$k$$-form on $$M$$ is a section of the $$k$$th exterior bundle

\bigwedge^k(M):= {\displaystyle \bigsqcup_{x \in M}} \bigwedge^k((T_x M)^*)

$$\pi$$

M

The space of all $$k$$-forms is denoted $$\Omega^k(M)$$.

If $$\omega \in \Omega^k(M)$$, then at each $$x \in M$$, $$\omega(x)$$ is an alternating multilinear map $$T_xM \times \dots \times T_x M \rightarrowtail \mathbb{R}$$.

Example. The standard area form on $$\mathbb{R}^2$$ is $$dx \wedge dy$$.

Example. The standard area form on $$S^2$$ is

\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}
x dy \wedge dz + y dz \wedge dx + z dx \wedge dy = d\theta \wedge dz = \sin \theta d\theta \wedge d\phi.

Think: alternating $$(0,k)$$-tensors

### Exterior Derivative

The exterior derivative is an anti-derivation $$d$$ of degree $$+1$$ that makes this a (co)chain complex.

0 \stackrel{d}{\to} \Omega^0(M) \stackrel{d}{\to} \Omega^1(M) \stackrel{d}{\to} \dots \stackrel{d}{\to} \Omega^{n-1}(M) \stackrel{d}{\to} \Omega^n(M) \to 0

In local coordinates

\displaystyle d\left( \sum_{i_1 < \dots < i_k} f_{i_1\dots i_k} dx_{i_1} \wedge \dots \wedge dx_{i_k}\right)
\displaystyle = \sum_{i_1 < \dots < i_k} \sum_{j=1}^n \frac{\partial f_{i_1 \dots i_k}}{\partial x^j} dx_j \wedge dx_{i_1} \wedge \dots \wedge dx_{i_k}.

de Rham’s Theorem. The cohomology of the above cochain complex agrees with the singular cohomology of $$M$$ with real coefficients.

### Symplectic Geometry

Definition. A symplectic form on a manifold $$M$$ is a closed, nondegenerate 2-form $$\omega$$. The pair $$(M,\omega)$$ is called a symplectic manifold.

Example. Any area form on an oriented surface; e.g., $$(\mathbb{R}^2,dx\wedge dy)$$, $$(S^2,d\theta\wedge dz)$$, $$(S^1 \times S^1, ds\wedge dt)$$, ...

Example. The canonical form $$\sum dq_i \wedge dp_i$$ on $$T^*\mathbb{R}^n$$ (or more generally any cotangent bundle).

Darboux’s Theorem. Every symplectic form looks like this locally.

### Consequences

A symplectic manifold must be even-dimensional (over $$\mathbb{R}$$).

$$\omega^{\wedge n} = \omega \wedge \dots \wedge \omega$$ is a volume form on $$M$$, and induces a measure

m(U) := \int_U \omega^{\wedge n}

called Liouville measure on $$M$$.

If $$H: M \to \mathbb{R}$$ is smooth, then there exists a unique vector field $$X_H$$ (the Hamiltonian vector field for $$H$$) so that $${dH = \iota_{X_H}\omega}$$, i.e.,

dH(\cdot) = \omega(X_H, \cdot)

($$X_H$$ is sometimes called the symplectic gradient of $$H$$)

### Example

$$H: (S^2, d\theta\wedge dz) \to \mathbb{R}$$ given by $$H(\theta,z) = z$$.

$$dH = dz = \iota_{\frac{\partial}{\partial \theta}}(d\theta\wedge dz)$$, so $$X_H = \frac{\partial}{\partial \theta}$$.

Integrating $$X_H$$ produces the one-parameter family of diffeomorphisms $$\psi_t(\theta, z) = (\theta+t,z)$$.

### Some Useful Non-Tensors

A (semi-)Riemannian metric $$g$$ on $$M$$ induces a unique connection called the Levi-Civita connection so that

Christoffel symbols $$\Gamma_{ij}^k$$ defined by $$\nabla_{X_i} X_j = \sum_k \Gamma_{ij}^k X_k$$

1. $$\nabla_U V - \nabla_V U = [U,V]$$ (symmetric)
2. $$U g(V,W) = g(\nabla_U V,W) + g(V, \nabla_U W)$$ (compatible)

Definition. An affine connection $$\nabla: \mathfrak{X}(M) \times \mathfrak{X}(M) \to \mathfrak{X}(M)$$ satisfies

1. $$\nabla_{f_1 V_1 + f_2V_2} W = f_1 \nabla_{V_1}W + f_2 \nabla_{V_2}W$$.
2. $$\nabla_V(W_1 + W_2) = \nabla_V W_1 + \nabla_V W_2$$.
3. $$\nabla_V f W = V(f) W + f \nabla_V W$$.

### Riemann Curvature Tensor

The Riemann curvature tensor is a $$(1,3)$$-tensor field

$$R: \mathfrak{X}(M) \times \mathfrak{X}(M) \to (\mathfrak{X}(M) \to\mathfrak{X}(M))$$ defined by

R(X,Y)Z=\nabla_Y \nabla_X Z -\nabla_X \nabla_Y Z + \nabla_{[X,Y]} Z

The sectional curvature of a 2-plane $$\sigma \subset T_x M$$ is

K(\sigma):=\frac{g(R(u,v)u,v)}{g(u,u)g(v,v)-g(u,v)^2},

where $$\{u,v\}$$ is any basis for $$\sigma$$.

At a point $$x \in M$$, the Ricci tensor $$\operatorname{Ric}: T_x M \times T_x M \to \mathbb{R}$$ is

\operatorname{Ric}(u,v) := \operatorname{tr}(w \mapsto R(w,v)u)

Or, in index notation, $$\operatorname{Ric}_{ij} = R_{ikj}^k = g^{k\ell}R_{ki\ell j}$$.

### Ricci Tensor

For unit $$u \in T_x M$$, the Ricci curvature of $$u$$ is the value of the corresponding quadratic form: $$\operatorname{Ric}(u,u)$$.

Fact. $$\operatorname{Ric}(u,u)$$ is $$n-1$$ times the average of $$K(\sigma)$$ for all 2-planes $$\sigma \subset T_x M$$ containing $$u$$.

The scalar curvature $$S = \operatorname{tr}_g \operatorname{Ric} = g^{ij}\operatorname{Ric}_{ij}$$

### Maxwell’s Equations

dF = 0
d\star F = \mu_0 J

(a 2-form)

current 3-form

### Einstein’s Field Equations

G=8\pi T

stress-energy tensor

Einstein tensor: $$G = \operatorname{Ric} - \frac{1}{2} S g$$

### Ricci Flow

g_t = -2 \operatorname{Ric}

### Grassmannians

Definition. The Grassmannian $$\operatorname{Gr}(d,V)$$ is the space of $$d$$-dimensional linear subspaces of the vector space $$V$$.

Example. $$\operatorname{Gr}(1,V) =\mathbb{P}(V)$$.

Example. $$\operatorname{Gr}(2,\mathbb{R}^4) \simeq (S^2 \times S^2)/(p,q)\sim (-p,-q)$$

### Plücker Embedding

\operatorname{Gr}(d,V) \to \mathbb{P}\left(\bigwedge^d V\right)
U \mapsto u_1 \wedge \dots \wedge u_d

$$\{u_1,\dots , u_d\}$$ a basis for $$U \subset V$$.

With respect to a preferred basis $$\{e_1, \dots , e_n\}$$ for $$V$$,

Theorem. The image is cut out by a system of quadratic equations (that say the wedge is decomposable).

Example. $$\operatorname{Gr}(2,\mathbb{K}^4)\mapsto\{\Delta_{12}\Delta_{34}-\Delta_{13}\Delta_{24}+\Delta_{14}\Delta_{23}=0\}$$

$$\Delta_U:=d!\, u_1 \wedge \dots \wedge u_d = \sum_{\sigma \in S_d} \operatorname{sgn}(\sigma) u_{\sigma(1)} \otimes \dots \otimes u_{\sigma(d)}$$ is an alternating $$d$$-tensor.

\displaystyle u_1 \wedge \dots \wedge u_d = \sum_{i_1 < \dots < i_d} \Delta_{i_1\dots i_d} e_{i_1} \wedge \dots \wedge e_{i_d}

where the $$\Delta_{i_1\dots i_d}$$ are the determinants of all the $$d \times d$$ minors of $$[u_1 \cdots u_d]$$.

### Projector Embedding

\operatorname{Gr}(d,\mathbb{R}^n) \to \operatorname{End}(\mathbb{R}^n)
\displaystyle U \mapsto \operatorname{Proj}_U

Proposition (with Chapman). Let $$V = \mathbb{R}^n$$. If $$\Delta_U$$ is constructed from an orthonormal basis for $$U$$, then

\Delta_U \cdot \Delta_U = (-1)^{d-1}(d-1)! \operatorname{Proj}_U.

$$\Delta_U(v)$$

$$\operatorname{Proj}_U(v)$$

$$v$$

$$\operatorname{Proj}_U$$ is a symmetric $$(1,1)$$-tensor.

{(\Delta_U)^k}_{i_1\dots i_{d-1}} {(\Delta_U)^{i_1\dots i_{d-1}}}_\ell = (-1)^{d-1}(d-1)! (\operatorname{Proj}_U)^k_\ell.

### Questions I Have

The flag mean of $$\{U_1,\dots , U_k\} \subset \operatorname{Gr}(d,\mathbb{R}^n)$$ is the span of the $$d$$ leading (left) singular vectors of $$\operatorname{Proj}_{U_1}+\dots + \operatorname{Proj}_{U_k}$$.

Question 1. Is there a notion of tensor SVD so that the flag mean of $$\{U_1,\dots , U_k\}$$ is the $$d$$ leading (left?) singular vectors of the $$\Delta_{U_1}+\dots+\Delta_{U_k}$$?

Question 2. What is the flag mean of the positive Grassmannian $$\operatorname{Gr}^+(d,\mathbb{R}^n)$$, consisting of those $$U$$ with all positive Plücker coordinates?

Question 3. Suppose $$U_1,U_2 \in \operatorname{Gr}(d,\mathbb{R}^n)$$ and you are given $$a \Delta_{U_1}+b \Delta_{U_2}$$ for unknown $$a,b \in \mathbb{R}$$. Can you determine $$U_1$$ and $$U_2$$?

Question 4. Is there a notion of Grassmannians of tensor subspaces? If so, what are good coordinates on this space?