The Shape of Space - Visualizing Surfaces & Three-Dimensional Manifolds

Question 1

What properties does traditional geometry have that topology lacks?

Answer 1

Curvature, areas, distances, and angles…basically all properties that change with respect to some sort of deformation of an object’s surface.

Question 2

What is the difference between Intrinsic vs Extrinsic properties?

Answer 2

In general, two surfaces have the same instrinsic topology if Flatlanders living in the surfaces cannot topologically tell one from the other.

Two surfaces have the same extrinsic topology if one can be deformed within three-dimensional space to look like the other.

For example, a Moebius Strip has the same intrinsic topology, but not the same extrinsic topology. A sheet of paper and the inner cardboard toilet paper roll has the same intrinsic topology but not the same extrinsic topology.

In this book we will study all surfaces intrinsically.

Question 3

The hemisphere, plane, and saddle surface all have…

same / different

…intrinsic geometries.

Answer 3

The hemisphere, plane, and saddle surface all have…

different

…intrinsic geometries.

Question 4

An intrinsically straight line is called a ________.

Answer 4

An intrinsically ​straight line is called a geodesic.

Question 5

What is the difference between local vs global properties?

Answer 5

Local properties are those observable within a small region of the manifold, whereas global properties require consideration of the manifold as a whole.

A flat torus and a doughnut surface have the same global topology but different local geometries. On the other hand, a flat torus and a plane have the same local geometry, but different global topologies.

A 2-dimensional manifold (ie - a surface) is a space with the local topology of a plane, and a 3-dimensional manifold is a space with the local topology of an ordinary 3D space.

Question 6

What is the difference between homogeneous vs non-homogenous manifolds?

Answer 6

A homogeneous manifold is one whose local geometry is the same at all points. (ex - a sphere, flat torus)

A non-homogeneous manifold varies its geometry from point to point. (ex - an irregular blob, or a doughnut since it is locally convex on the outside but concave on the inside)

Question 7

Give an example of a 1D manifold.

Answer 7

One-dimensional manifolds include lines and circles, but not figure eights (because no neighborhood of their crossing point is homeomorphic to Euclidean 1-space).

Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, which can all be embedded (formed without self-intersections) in three dimensional real space, but also the Klein bottle and real projective plane, which will always self-intersect when immersed in three-dimensional real space

Question 8

Can we find a surface in tridimensional space representing the square flat torus without length distortion?

What is the name of the term used to describe this process

Answer 8

There does not exist any way to embed the square flat torus in tridimensional space while preserving distances.

We say that the square flat torus does not admit any isometric embedding in the ambient space.

Question 9

Which of the following manifolds are closed vs. open?

(Corresponding to “finite” vs. “infinite”,

or “has an edge” vs. “no edge”)

1. Circle
2. Line
3. Two-hole donut surface
4. Sphere
5. Plane
6. Infinitely long cylinder
7. Flat Torus
8. Three-Dimensional Space
9. Three-Torus

Answer 9

Which of the following manifolds are closed vs. open?

1. Circle = Closed
2. Line = Open
3. Two-hole donut surface = Closed
4. Sphere = Closed
5. Plane = Open
6. Infinitely long cylinder = Open
7. Flat Torus = Closed
8. Three-Dimensional Space = Open
9. Three-Torus = Closed

Question 10

The local geometry of a Klein bottle is _________.

Answer 10

The local geometry of a Klein bottle is everywhere flat.

Question 11

A path in a surface or 3-manifold which brings a traveler back to his starting point mirror-reversed is called an ________-________ _______.

Answer 11

A path in a surface or 3-manifold which brings a traveler back to his starting point mirror-reversed is called an orientation-reversing path.

Question 12

Manifolds that don’t contain orientation-reversing paths are called ______.

Manifolds that do contain orientation-reversing paths are ______.

Answer 12

Manifolds that don’t contain orientation-reversing paths are called orientable.

Manifolds that do contain orientation-reversing paths are non-orientable.

Question 13

Name 2 examples of orientable surfaces.

Name an example of a non-orientable surface.

Answer 13

Orientable Surfaces: sphere, torus

Non-orientable Surface: Klein bottle

Question 14

The projective plane is a surface that is globally / locally like a sphere, but has same / different global topology.

How do you make a projective plane?

Answer 14

The projective plane is a surface that is locally like a sphere, but has different global topology.

You can make a projective plane by gluing together the opposite points on the rim of a hemisphere.

Question 15

The projective plane is homogeneous / non-homogeneous.

Answer 15

The projective plane is homogeneous.

Question 16

Label the following with orientable vs. non-orientable and curved local geometry vs. flat local geometry:

1. Sphere
2. Torus
3. Klein Bottle
4. Projective Plane

Answer 16

Label the following with orientable vs. non-orientable and curved local geometry vs. flat local geometry:

1. Sphere = Orientable + Curved
2. Torus = Orientable + Flat
3. Klein Bottle = Non-Orientable + Flat
4. Projective Plane = Non-Orientable + Curved

Question 17

Is Orientability a local or global property?

Is it topological or geometrical?

Answer 17

Orientability is a global property because it says something about a manifold as a whole.

It is a topological property because deforming a manifold does not affect it.

Question 18

The operation of making a two-holed donut surface from two one-holed ones by cutting a disk out of each and gluing together the exposed edges is called a _______-____.

Answer 18

The operation of making a two-holed donut surface from two one-holed ones by cutting a disk out of each and gluing together the exposed edges is called a connected-sum.

Question 19

What do you get when you form the connected sum of a two-holed donut surface and a sphere?

How about a Klein bottle and a sphere?

A projective plane and a sphere?

Answer 19

What do you get when you form the connected sum of a two-holed donut surface and a sphere?

Two-holed donut surface

How about a Klein bottle and a sphere?

Klein Bottle

A projective plane and a sphere?

Projective Plane

Question 20

What do you get when you cut a disk out of a projective plane?

What is the connected sum of two projective planes?

Answer 20

What do you get when you cut a disk out of a projective plane?

Mobius Strip

What is the connected sum of two projective planes?

Klein Bottle

Question 21

What are the shorthand names for the following simple manifolds?

1. euclidean plane
2. sphere
3. torus
4. klein bottle
5. projective plant
6. disk

Answer 21

What are the shorthand names for the following simple manifolds?

1. euclidean plane = E² = (“E-two”)
2. sphere = S²
3. torus = T² (both donut + flat)
4. klein bottle = K²
5. projective plant = P²
6. disk = D²

Question 22

What are the shorthand names for the following manifolds?

1. euclidean space
2. three-torus
3. solid ball
4. projective three-space

Answer 22

What are the shorthand names for the following manifolds?

1. euclidean space = E³
2. three-torus = T³
3. solid ball (a 3D “disk”) = D³
4. projective three-space = P³

Question 23

What are the shorthand names for the following manifolds?

1. line
2. circle
3. interval

Answer 23

What are the shorthand names for the following manifolds?

1. line = E¹
2. circle = S¹
3. interval = I

Question 24

The connected sum operation is abbreviated with the ___-symbol.

Answer 24

The connected sum operation is abbreviated with the ”#” symbol.

Question 25

How do you notate the two-holed donut surface as a connected sum? \_\_\_\_\_ How do you notate the three-holed donut surface as a connected sum?

Answer 25

How do you notate the two-holed donut surface as a connected sum? **T² # T²** Since it is the connected sum of two tori. \_\_\_\_ How do you notate the three-holed donut surface as a connected sum? **T² # T²# T²**

Question 26

Sometime in the 1860s, mathematicians have discovered that *every* conceivable surface is a connected sum of _____ and/or \_\_\_\_\_. What is the sphere a connected sum of?

Answer 26

Sometime in the 1860s, mathematicians have discovered that *every*​ conceivable surface is a connected sum of **tori** and/or **projective planes**. \_\_\_\_\_ What is the sphere a connected sum of? **zero tori and zero projective planes**

Question 27

What is a Klein bottle a connected sum of?

Answer 27

What is a Klein bottle a connected sum of? ## Footnote **P² # P²**

Question 28

Simplify: K² # P² = K² # T² ​= K² # K² ​=

Answer 28

Simplify: K² # P² = **P² # P² # P²** K² # T² ​= **P² # P² # ​T²** K² # K² ​= **P² # P² # P² # P²**

Question 29

Simplify: K² # P² ​= T² # P² =

Answer 29

Simplify: K² # P² ​= **P² # P² # ​P²** T² # P² = **K² # P² = P² # P² # ​P²**

Question 30

Every surface is a connected sum of either ____ *only* or ____ *only!!!*

Answer 30

Every surface is a connected sum of either **tori** *only* or **projective planes***​ only!!!*

Question 31

A cylinder is the *product* of a _____ and an \_\_\_\_\_\_. This is notated as \_\_\_\_\_.

Answer 31

A cylinder is the *product* of a **circle** and an **interval**. This is notated as **S¹ x I**

Question 32

A circle is the *product* of ____ and a \_\_\_\_\_. This is notated as \_\_\_\_\_.

Answer 32

A circle is the *product* ​of **circle** and a **circle**. This is notated as **S¹ x S¹** (btw, the torus is the only closed surface (with no edges) that is a product!!! The reason is that a two-dimensional thing must be the product of two one-dimensional things, and the circle is the only one-dimensional thing available that has neither end-points (like an interval) nor infinite length (like a line), thus S¹ x S¹ is the only two-dimensional product having neither an edge nor an infinite area!!!)

Question 33

What are the usual names for the following products: 1. I x I 2. E¹ x E¹ 3. S¹ x E¹ 4. E¹ x I

Answer 33

What are the usual names for the following products: 1. I x I = **Square** 2. E¹ x E¹ = **2D** **Euclidean Plane** 3. S¹ x E¹ = **Infinite** **Cylinder** 4. E¹ x I = **Infinite Strip**

Question 34

Is the Mobius strip a product?

Answer 34

No. It is a circle of intervals, but not an interval of circles! (due to the non-orientability)

Question 35

What's D² x S¹?

Answer 35

What's D² x S¹? ## Footnote **Solid Torus**

Question 36

Is the cylinder a product in the topological sense? How about the geometric sense?

Answer 36

Is the cylinder a product in the topological sense? **Yes** How about the geometric sense? **Yes, because it satisfies the following three properties:** **1) All circles are the same size** **2) All intervals are the same size** **3) Each circle is perpendicular to each interval**

Question 37

Draw one version of I x I that is a geometric product. Draw one version that is *only* a topological product.

Answer 37

Draw one version of I x I that is a geometric product: **A square** Draw one version that is *only* a topological product: **Any *deformed* version of a square**

Question 38

What is the geometrical product of S¹ x S¹?

Answer 38

What is the geometrical product of S¹ x S¹? ## Footnote **flat torus**

Question 39

The three-torus is a product of...

Answer 39

The three-torus is a product of a **two-torus** and a **circle.** In other words a *circle of tori* or a *torus of circles.* **T³ = T² x S¹** This is because a three-torus can be imagined as a cube with the opposite sides glued. Imagine this cube to consist of a stack of horizontal layers. When the cube's sides get glued, each horizontal later gets converted into a torus (a flat torus to be specific). At this stage we have a stack of flat tori. When the cube's top is glued to its bottom, this stack of tori is converted into a circle of tori. Next we have to check the commutativity - that it is a torus of circles as well. And this three-torus is a geometric product as well because all horizontal tori are the same size, all vertical circles are the same size, and each torus is perpendicular to each circle.

Question 40

Imagine what the following manifolds might look like: S² x I S² x S¹

Answer 40

Imagine what the following manifolds might look like: S² x I **This is like a volumetric ball with a hollow center** S² x S¹ **In other words, it is a *sphere* *cross circle*.** **It looks like the previous shape but where the inner surface in the core is glued to the outer surface of the ball. This converts the horizontal ring-like surface into a torus.**

Question 41

A ________ manifold is one whose local geometry is *everywhere the same*. A _______ manifold is one in which the geometry is the *same in all directions*.

Answer 41

A **homogeneous** manifold is one whose local geometry is *everywhere the same*. A **isotropic** manifold is one in which the geometry is the *same in all directions*.

Question 42

S² x S¹ is an example of a three-manifold that is **homogeneous / non-homogeneous** and **isotropic / non-isotropic**.

Answer 42

S² x S¹is an example of a three-manifold that is **homogeneous** and **non-****isotropic.** It is not isotropic because some two-dimensional slices have the local geometry of a sphere, while other slices have the local geometry of the plane.

Question 43

Find a non-orientable three-manifold that is a product and has the same local geometry as S² x S¹. Hint: What might such a manifold be the product of?

Answer 43

Find a non-orientable three-manifold that is a product and has the same local geometry as S² x S¹. **P² x S¹** You can visualize this as a thickened hemisphere. The inside surface is glued to the outside surface and the points on the "rim surface" are glued so that each hemispherical layer becomes a projective plane. This manifold is locally identical to S² x S¹, but its global topology is different.

Question 44

Do a hexagonal torus and an ordinary flat torus have the same local topology? The same local geometry? Same global topology? Same global geometry?

Answer 44

Do a hexagonal torus and an ordinary flat torus have the same local topology? **Yes** The same local geometry? (flat or euclidean at all points) **Yes** Same global topology? **Yes** Same global geometry? **No**

Question 45

The _____ \_\_\_\_\_ can conclusively decide whether or not two *surfaces* are the same topologically.

Answer 45

The **Euler number** can conclusively decide whether or not two *surfaces* are the same topologically.

Question 46

Is **sidedness** an intrinsic or extrinsic property of a surface? What about orientability? Are all Klein bottles orientable? Are they all one-sided?

Answer 46

Is **sidedness** an intrinsic or extrinsic property of a surface? **extrinsic** What about orientability? **intrinsic** Are all Klein bottles orientable? **no** Are they all one-sided? **no, only some are**

Question 47

A triangle drawn on a sphere is called a ______ \_\_\_\_\_\_.

Answer 47

A triangle drawn on a sphere is called a **spherical triangle**.

Question 48

Each side of a *spherical triangle* is required to be a \_\_\_\_\_\_; that is, it is required to be *intrinsically straight* in the sense that a Flatlander on the sphere would perceive it bending neither to the left nor right.

Answer 48

Each side of a *spherical triangle* is required to be a **geodesic**; that is, it is required to be *intrinsically straight* in the sense that a Flatlander on the sphere would perceive it bending neither to the left nor right.

Question 49

The *unit sphere* has an area of \_\_\_\_.

Answer 49

The *unit sphere* has an area of **4π**.

Question 50

What is the area of a spherical triangle whose angles in radians are π/2, π/3, and π/4?

Answer 50

What is the area of a spherical triangle whose angles in radians are π/2, π/3, and π/4? ## Footnote **The sum of the angles of the first triangle is π​/2 + π/3 + π/4 = 13π/12, so its area must be 13π/12 - π = π/12** **(where π is the area of half of a hemisphere)**

Question 51

The region on a sphere bounded by the double intersection of 2 great circles is called a ______ \_\_\_\_\_\_.

Answer 51

The region on a sphere bounded by the double intersection of 2 great circles is called a **double lune**.

Question 52

The largest the angle a can ever be in a double lune is \_\_\_.

Answer 52

The largest the angle a can ever be in a double lune is **π**. (at which point the double lune fills up the entire sphere)

Question 53

So if alpha = π/3, then since π/3 is 1/3 the greatest possible angle π, we thus know that the double lune must fill up 1/3 the area of the entire sphere, namely (1/3)\*(4π) = 4π/3. Using the same reason, we can generally say that given angle a, what is the area of the double lune?

Answer 53

So if alpha = π/3, then since π/3 is 1/3 the greatest possible angle π, we thus know that the double lune must fill up 1/3 the area of the entire sphere, namely (1/3)\*(4π) = 4π/3. Using the same reason, we can generally say that given angle a, what is the area of the double lune? **(a/π)(4π) = 4a**

Question 54

Given angle a, we know that the area of a double lune is 4a. Using this reasoning, what is the area of a triangle on a sphere with angles a, b, and c?

Answer 54

Given angle a, we know that the area of a double lune is 4a. Using this reasoning, what is the area of a triangle on a sphere with angles a, b, and c? **We know that each double lune has an area of 4a, 4b, and 4c respectively. We also know that to make a triangle on a sphere we can overlap three great circles to get:** **(area of first double lune) + (area of second double lune) + (area of third double lune) = (everything shaded in once) - (each triangle shaded in two more times)** **= (area of entire sphere) + 2(area of original triangle) + 2(area of antipodal triangle)** **= 4a + 4b + 4c = 4π + 2A + 2A** **= 4(a + b + c) = 4(****π + A)** **= (a + b + c) - π = A** **Thus, the sum of the angles of a spherical triangle exceeds π by an amount equal to the triangle's area.**

Question 55

What is one way that the geometry of a sphere differs from the geometry of a plane? Can you represent this mathematically?

Answer 55

What is one way that the geometry of a sphere differs from the geometry of a plane? **The sum of angles of a spherical triangle exceeds π by an amount proportional to the triangle's area, whereas the sum of the angles of a Euclidean (= flat) triangle equals π exactly.** Can you represent this mathematically? **= (a + b + c) - π = A**

Question 56

What kind of *curvature* do the following geometries have? 1. elliptic geometry 2. euclidean geometry 3. hyperbolic geometry

Answer 56

What kind of *curvature* do the following geometries have? 1. elliptic geometry **= positive curvature** 2. euclidean geometry **= zero curvature** 3. hyperbolic geometry **= negative curvature**

Question 57

The *hyperbolic plane (H²)* is an infinite plane that has hyperbolic geometry or _______ \_\_\_\_\_\_\_ curvature everywhere.

Answer 57

The *hyperbolic plane (H²)* is an infinite plane that has hyperbolic geometry or **constant** **negative** curvature everywhere.

Question 58

Draw a picture showing how to cut a four-holed surface (T²#T²#T²#T²) whose corners meet in groups of four. Do the same for a two-holed donut surface (T²#T²). How many hexagons do you get when you cut up an *n-*holed donut surface?

Answer 58

Draw a picture showing how to cut a four-holed surface (T²#T²#T²#T²) whose corners meet in groups of four. Do the same for a two-holed donut surface (T²#T²). How many hexagons do you get when you cut up an *n-*holed donut surface? **For a two-holed surface you get 4 hexagons. For a 4 holed surface you get 12 hexagons.** **When you cut an *n*-holed surface into hexagons whose corners meet in groups of four, you get:** **= 4n-4 hexagons**

Question 59

Any donut surface with at least two holes can be cut into _______ whose corners meet in groups of four. Thus, any such surface can be given a _______ geometry.

Answer 59

Any donut surface with at least two holes can be cut into **hexagons** whose corners meet in groups of four. Thus, any such surface can be given a **hyperbolic** geometry. (By way of homogeneously increasing the angles of a hexagon and then embedding it onto a two-dimensional surface. That surface will have to "buckle" and cannot be flattened, and hence will have a hyperbolic geometry) **Thus, every surface can be given some type of homogeneous geometry.**

Question 60

The connected sum of two projective planes can be cut into ___ \_\_\_\_\_\_. Similarly, P²#P²#P² can be cut into into ___ \_\_\_\_\_\_. Similarly, P²#P²#P²#P² can be cut into ___ \_\_\_\_\_\_\_. Which of these surfaces can be given which homogeneous geometry? (elliptic, euclidean, or hyperbolic)

Answer 60

The connected sum of two projective planes can be cut into **2 squares**. Similarly, P²#P^2#P² can be cut into **2 hexagons**. Similarly, P²#P^2#P²#P²​ can be cut into **2 octagons**. Which of these surfaces can be given which homogeneous geometry? (elliptic, euclidean, or hyperbolic) **P² has elliptic geometry to begin with. So...** P²#P² = **euclidean geometry (since the 90º corners of flat squares fit nicely in groups of 4, and we also know P²#P²=K² and that has euclidean geometry as well)** **All other cases the polygons' corners must be shrunk to fit together in groups of four, so all the other connected sums of projective planes can be given hyperbolic geometry.** P²#P²#P² = **hyperbolic geometry** P²#P²#P²#P² = **hyperbolic geometry**

Question 61

The connective sum of n projective planes can be divided up into ____ 2n-gons whose corners meet in groups of four.

Answer 61

The connective sum of n projective planes can be divided up into **two** 2n-gons whose corners meet in groups of four.

Question 62

Group the following surfaces into their corresponding *geometry* and *orientability* : 1. S² 2. P² 3. T² 4. P²#P² = K² 5. T²#T²#... 6. P²#P²#...

Answer 62

Group the following surfaces into their corresponding *geometry* and *orientability* : 1. S² = **elliptic + orientable** 2. P² = **elliptic + non-orientable** 3. T² = **euclidean** **+ orientable** 4. P²#P² = K² = **euclidean** **+ non-orientable** 5. T²#T²#... = **hyperbolic** **+ orientable** 6. P²#P²#... = **hyperbolic** **+ non-orientable**

Question 63

Every surface has an _____ \_\_\_\_\_, which is an integer that contains essential information about the surface's *global topology*.

Answer 63

Every surface has an **Euler Number**, which is an integer that contains essential information about the surface's *global topology*.

Question 64

Surfaces with a positive Euler number have a _____ geometry. If they have a zero Euler number they have a ______ geometry. If they have a negative Euler number they have a _____ geometry.

Answer 64

Surfaces with a positive Euler number have a **elliptic** geometry. If they have a zero Euler number they have a **Euclidean** geometry. If they have a negative Euler number they have a **hyperbolic** geometry.

Question 65

A zero-dimensional cell is a ______ or \_\_\_\_\_. A one-dimensional cell is topologically a ____ \_\_\_\_\_\_ or an \_\_\_\_\_. A two-dimensional cell is a ______ or a \_\_\_\_\_. A ___ \_\_\_\_\_\_\_ is what you get when you divide a surface into cells (like a decomposition of a surface into polygons).

Answer 65

A zero-dimensional cell is a **point** or **vertex**. A one-dimensional cell is topologically a **line segment** or an **edge**. A two-dimensional cell is a **polygon** or a **face**. A **cell division** is what you get when you divide a surface into cells (like a decomposition of a surface into polygons).

Question 66

How many vertices, edges, and faces are in a cell division of S²?

Answer 66

How many vertices, edges, and faces are in a cell division of S²​? ## Footnote **V = 6** **E = 12** **F = 8**

Question 67

How many vertices, edges, and faces are in a cell division of P²​?

Answer 67

How many vertices, edges, and faces are in a cell division of P^2​? ## Footnote **V = 9** **E = 15** **F = 7** **(vertices and edges on the disk are glued together to their corresponding opposite side pairs)**

Question 68

Split up T²#T² into cellular divisions​​ of polygons.

Answer 68

Split up T²#T² into cellular divisions​​ of polygons. **This is a torus glued to another torus. You can cut this up into 2 octagons + 4 squares OR 2 hexagons.**

Question 69

We know that the area of a triangle on a unit sphere is the sum of its angles minus π: A = (a + b + c) - π We also know that the area of an n-gon is the sum of the areas of n-2 triangles. What is the area of an n-gon on a unit sphere?

Answer 69

We know that the area of a triangle *on a unit sphere* is the sum of its angles minus π: A_triangle = (a + b + c) - π We also know that the area of an n-gon is the sum of the areas of n-2 triangles (because we can divide a polygon into n-2 constituent triangles) Imagine dividing up a pentagon into triangles. What is the area of an n-gon on a unit sphere? **A_n-gon = (sum of all angles) - (n-2)π** **You can interpret this as saying that the area of a spherical n-gon equals the difference between what the angles actually are and what they would have been if the n-gon were flat. ie - if you took a bowl and squashed it flat, we are subtracting out the influence of the angles.**

Question 70

What is the *Unified Gauss-Bonnet Formula* for surfaces of constant curvature?

Answer 70

What is the *Unified Gauss-Bonnet Formula* ​for surfaces of constant curvature? ## Footnote **kA = 2πx** **where** **k = -1 (hyperbolic curvature), 0 (euclidean), +1 (elliptic)** **A = surface area** **x = Euler number**

Question 71

The Gauss-Bonnet formula can be generalized to apply to \_\_\_-\_\_\_\_\_\_\_ surfaces whose Gaussian curvature varies irregularly from point to point.

Answer 71

The Gauss-Bonnet formula can be generalized to apply to **non-homogeneous** surfaces whose Gaussian curvature varies irregularly from point to point. ## Footnote **The idea is that the + and - curvature cancel and the net total curvature equals 2πx. On a donut surface, the positive curvature cancels the negative curvature exactly. (The outer, convex portion of the donut surface has + curvature, and the portion in the hole is - curvature). Whenever you create positive curvature in one place, you create an equal amount of negative curvature in another place!**

Question 72

Whenever you create positive curvature in one place, you create an *equal / greater / less* amount of negative curvature in another place!

Answer 72

Whenever you create positive curvature in one place, you create an **equal** amount of negative curvature in another place!

Question 73

Using the language of integral calculus, state the Gauss-Bonnet formula precisely.

Answer 73

Using the language of integral calculus, state the Gauss-Bonnet formula precisely: ## Footnote **= Σ k•A = = 2πx** **= Integral of k • dA = 2πx** **...and when k is constant, this reduces to...** **kA = 2πx**

Question 74

The ________ \_\_\_\_\_ is a tetrahedron with each 2 adjacent faces glued together. Do the corners have to expand or shrink to fit properly? What homogeneous geometry does this manifold admit?

Answer 74

The **tetrahedral space** is a tetrahedron with each 2 adjacent faces glued together. Do the corners have to expand or shrink to fit properly? **The corners are too pointy to fit together snugly, so the tetrahedron must be expanded in a hypersphere.** What homogeneous geometry does this manifold admit? **This gives the manifold an elliptic geometry.**

Question 75

The ________ \_\_\_\_\_\_ is a cube with each face glued to the opposite face with one-quarter clockwise turn. How do the cube's corners fit together? What homogeneous geometry does this manifold admit?

Answer 75

The ________ \_\_\_\_\_\_ is a cube with each face glued to the opposite face with one-quarter clockwise turn. How do the cube's corners fit together? **The cube's corners come together in two groups of four corners each. Each corner is adjacent to the other three in its group, but to no others.** What homogeneous geometry does this manifold admit? **The corners are *too small* to fit snugly in groups of four, and the manifold ends up with an *elliptic geometry*.** **Btw, the manifold's name arises from the fact that its symmetries can be modeled in the quaternions, a number system like the complex numbers but with three imaginary quantities instead of one. It also lacks commutativity.**

Question 76

If a geometry is \_\_\_\_\_\_\_, but not \_\_\_\_\_\_, then we know that it's everywhere the same, but at any given point we cannot distinguish some directions from others.

Answer 76

If a geometry is **homogeneous**, but not **isotropic**, then we know that it's everywhere the same, but at any given point we cannot distinguish some directions from others.

Question 77

The term _______ \_\_\_\_\_\_\_ refers to the curvature of a two-dimensional slice of a manifold. An _______ geometry has the same sectional curvature in all directions.

Answer 77

The term **sectional curvature** refers to the curvature of a two-dimensional slice of a manifold. An **isotropic** geometry has the same sectional curvature in all directions. **ie - the sectional curvatures of 3D elliptic geometry are all positive, those of 3D Euclidean geometry are all zero, and those of 3D hyperbolic geometry are all positive**

Question 78

*Only* homogeneous \_\_-dimensional geometries are elliptic, euclidean, or hyperbolic, all of which are isotropic. Geometries that are homogeneous but not isotropic occur only in manifolds of ____ or more dimensions.

Answer 78

*Only* homogeneous **2**-dimensional geometries are elliptic, euclidean, or hyperbolic, all of which are isotropic. Geometries that are homogeneous but not isotropic occur only in manifolds of **3** or more dimensions.

Question 79

Research by Bill Thurston suggests that three-dimensional ________ geometry is by far the most common geometry for three-manifolds. And two-dimensional ________ geometry is the most common geometry for surfaces!

Answer 79

Research by Bill Thurston suggests that three-dimensional **hyperbolic geometry** is by far the most common geometry for three-manifolds. And two-dimensional **hyperbolic geometry** is the most common geometry for surfaces! (see page 249 of book)