Riemannian Geometry_Part 2_Riemannian metrics_John Lee_Introduction to Riemannian Manifolds p01 11_Riemannian metrics

1. Formula
   Observation

   Written as V in the definition of linear isometry.

V

An inner product space.

Inner product space

1. Formula
   Observation

   Written as W in the definition of linear isometry.

W

Another inner product space.

Inner product space

1. Formula
   Observation

   Written as F: $V \to W$.

F

A map between two inner product spaces.

Map from V to W

1. Formula
   Observation

   Written as a, b in the inner product preservation equation.

a, b

Arbitrary vectors in V.

Vectors in V

1. Formula
   Observation

   Written as k in the linearity condition F(ka) = kF(a).

k

A scalar in the underlying field.

Field

1. Formula
   Observation

   The handwritten text reads "Let V, W be inner product spaces."

V

An inner product space.

Inner product space

1. Formula
   Observation

   The handwritten text reads "Let V, W be inner product spaces."

W

An inner product space.

Inner product space

1. Formula
   Observation

   The handwritten text reads "A map F: V -> W is called a linear isometry..."

F

A map between inner product spaces.

Map from V to W

1. Formula
   Observation

   The handwritten text reads "Let M be a smooth manifold."

M

A smooth manifold.

Smooth manifold

1. Formula
   Observation

   The handwritten text reads "A Riemannian metric g on M is a smooth covariant 2-tensor field..."

g

A Riemannian metric on M.

Smooth covariant 2-tensor field on M

1. Formula
   Observation

   The handwritten text reads "(in $T^2(M)$)".

$T^2(M)$

The space of covariant 2-tensor fields on M.

Space of tensor fields

1. Formula
   Observation

   The handwritten text reads "whose value $g_p$ at $p \in M$..."

p

A point on the smooth manifold M.

Point in M

1. Formula
   Observation

   Text on screen: 'Let V, W be inner product spaces. A map F: $V \to W$ is called a linear isometry if it is a vector space isomorphism that preserves inner products.'

2. Audio
   Observation

   Speaker reads the definition and explains the conditions.

A linear isometry is a map between two inner product spaces that is both a vector space isomorphism and preserves inner products.

1. V and W are inner product spaces.

2. F is a vector space isomorphism.

3. F preserves inner products for all a, $b \in V$.

1. Formula
   Observation

   Written equations: $F(a+b) = F(a) + F(b)$ and F(ka) = kF(a).

2. Audio
   Observation

   Speaker states these are the conditions for being a vector space isomorphism.

To be a vector space isomorphism, the map must preserve vector addition and scalar multiplication.

1. a, $b \in V$

2. k is a scalar in the field

1. Formula
   Observation

   The handwritten text defines a linear isometry: "Let V, W be inner product spaces. A map F: V -> W is called a linear isometry if it is a vector space isomorphism that preserves inner products."

A linear isometry is defined as a map between two inner product spaces that is both a vector space isomorphism and preserves inner products.

1. V and W are inner product spaces.

2. F is a vector space isomorphism.

3. F preserves inner products.

1. Formula
   Observation

   The handwritten text defines a Riemannian metric: "Let M be a smooth manifold. A Riemannian metric g on M is a smooth covariant 2-tensor field (in $T^2(M)$), whose value $g_p$ at $p \in M$ is an inner product on T_pM. Thus g is a symmetric 2-tensor field that is positive definite in the sense that $g_p(v,v) \ge 0$, $g_p(v,v)=0 \iff v=0$."

2. Audio
   Observation

   The speaker explains that a Riemannian metric is a smooth covariant 2-tensor field and elaborates on the meaning of 'covariant' and 'tensor'.

A Riemannian metric on a smooth manifold is a smooth covariant 2-tensor field whose value at each point is an inner product on the tangent space at that point. It is symmetric and positive definite.

1. M is a smooth manifold.

2. g is a smooth covariant 2-tensor field.

3. For each $p \in M$, $g_p$ is an inner product on T_pM.

4. $g_p(v,v) \ge 0$ for all $v \in T$_pM.

5. $g_p(v,v) = 0$ if and only if $v = 0$.

1. Linear Isometry

1. Formula
   Observation

   The handwritten definition of a Riemannian metric is visible throughout the clip.

A Riemannian metric g on a smooth manifold M is defined as a smooth covariant 2-tensor field whose value at each point p is an inner product on the tangent space T_pM. It must be symmetric and positive definite.

1. M is a smooth manifold

2. g is a smooth covariant 2-tensor field

3. $g_p$ is an inner product on T_pM for all $p \in M$

1. Audio
   Observation

   The speaker explains that a tensor field allows coefficient functions to be pulled out.

2. Formula
   Observation

   The speaker writes the equation g(fX, hY) = fh $g(X,Y)$.

For a covariant 2-tensor field g, it is linear over the ring of smooth functions. This means that if you multiply the input vector fields by smooth functions, those functions can be factored out of the tensor evaluation.

1. g is a covariant 2-tensor field

2. X, Y are vector fields

3. f, h are smooth functions

1. Definition of Riemannian Metric

1. Audio
   Observation

   The speaker states that 'smooth' means locally in a chart it is smooth in Euclidean space.

The smoothness of a tensor field like a Riemannian metric is defined locally. In any local coordinate chart, the component functions of the tensor must be smooth (infinitely differentiable) as functions on Euclidean space.

1. Applicable in local coordinates

1. Definition of Riemannian Metric

1. Formula
   Observation

   The speaker writes TM = $\bigsqcup_{p \in M} T$_pM.

2. Audio
   Observation

   The speaker defines the tangent bundle as the disjoint union of all tangent spaces.

The tangent bundle TM of a manifold M is the disjoint union of all tangent spaces T_pM for every point p in M. It naturally inherits a smooth manifold structure from M.

1. M is a smooth manifold

1. Definition of Riemannian Metric

1. Formula
   Observation

   The board states: "Let M be a smooth manifold. A Riemannian metric g on M is a smooth covariant 2-tensor field (in $\mathcal{T}^2(M)$), whose value $g_p$ at $p \in M$ is an inner product on T_pM. Thus g is a symmetric 2-tensor field that is positive definite in the sense that $g_p(v,v) \ge 0$, $g_p(v,v)=0 \Leftrightarrow v=0$."

2. Audio
   Observation

   The lecturer explains that g is symmetric because its restriction to each p is an inner product on the tangent space, and adds that it is positive definite because $g_p(v,v)$ is nonnegative and equals zero iff $v=0$.

A Riemannian metric on a smooth manifold M is defined as a smooth covariant 2-tensor field g such that at every point $p \in M$, the value $g_p$ is an inner product on the tangent space T_pM. The lecturer unpacks this by saying that g must be symmetric and positive definite pointwise.

1. M is a smooth manifold.

2. g is a smooth covariant 2-tensor field.

3. For each $p \in M$, $g_p$ is an inner product on T_pM.

4. Symmetry is stated verbally as $g(x,y)=g(y,x)$.

5. Positive definiteness is stated pointwise on tangent vectors $v \in T$_pM.

1. Formula
   Observation

   The board writes: "Endowed with a Riemannian metric, (M,g) is called a Riemannian manifold."

2. Audio
   Observation

   The lecturer says that when a smooth manifold M is endowed with a Riemannian metric, one calls this a Riemannian manifold, and that this is the object studied in the course.

Once a smooth manifold M is equipped with a Riemannian metric g, the pair (M,g) is called a Riemannian manifold. This is the main object of study introduced in the clip.

1. M is a smooth manifold.

2. g is a Riemannian metric on M.

1. Definition of a Riemannian metric

1. Formula
   Observation

   The board shows the heading "[Prop, Existence of Riemannian metric]" and the statement "Every smooth manifold admits a Riemannian metric."

2. Audio
   Observation

   The lecturer says, "In the next part we will study the existence of Riemannian metrics."

3. Formula
   Observation

   The lower page begins the proof with "Rmk: Use a partition of unity." followed by an atlas and subordinate partition of unity.

1. Only the beginning of the proof is visible before the clip ends; the full argument is not contained in this segment.

The clip introduces the proposition that every smooth manifold admits a Riemannian metric and begins the proof strategy by saying to use a partition of unity. The visible setup chooses an atlas $\mathcal{A}$ of M and a partition of unity \{This formula needs review. Please report it using the page feedback control.alpha\} subordinate to it, then defines a local tensor This formula needs review. Please report it using the page feedback control.alpha = $\sum_i$ du_\alpha^$i \otimes$ du_\alpha^i on each chart.

1. M is a smooth manifold.

2. The proof uses an atlas $\mathcal{A}$=\{$F_\alpha$: $U_\alpha \to Q_\alpha \subseteq \mathbb{R}^n$\}.

3. It also uses a partition of unity \{This formula needs review. Please report it using the page feedback control.alpha: $U_\alpha \to [0,1]$\} subordinate to $\mathcal{A}$.

1. Definition of a Riemannian metric

1. Formula
   Observation

   The handwritten text explicitly states the conditions for positive definiteness.

A Riemannian metric g is positive definite, meaning for any tangent vector v, $g_p(v,v) \ge 0$, and $g_p(v,v) = 0$ if and only if $v = 0$.

1. g is a Riemannian metric on M

2. $v \in T$_pM

$\forall p \in M$, $\forall v \in T$_pM

1. Formula
   Observation

   The handwritten text describes g as a 'symmetric 2-tensor field'.

A Riemannian metric g is a symmetric 2-tensor field, implying $g(X,Y) = g(Y,X)$ for any vector fields X and Y.

1. g is a Riemannian metric on M

2. X, Y are vector fields on M

$\forall X$, $Y \in \mathfrak{X}(M)$

1. Formula
   Observation

   The board explicitly states: "Every smooth manifold admits a Riemannian metric."

2. Audio
   Observation

   The lecturer announces that the next part will study the existence of Riemannian metrics.

1. The proof is only initiated in this clip and is not completed within the provided duration.

Every smooth manifold admits a Riemannian metric.

1. M is a smooth manifold.

For every smooth manifold M, there exists a Riemannian metric g on M.

1. Formula
   Observation

   Written equation: $\langle a$, $b \rangle_V = \langle F(a)$, $F(b) \rangle_W$.

2. Audio
   Observation

   Speaker explains that picking two vectors a and b and taking their inner products in V equals taking the inner product of their images in W.

1. Expression

   $$\langle a, b \rangle_V$$
   Explanation

   Take the inner product of two arbitrary vectors a and b in the domain space V.

   Justification

   Definition of inner product space.

   Shown in the video
2. Expression

   $$\langle F(a), F(b) \rangle_W$$
   Explanation

   Take the inner product of their mapped images $F(a)$ and $F(b)$ in the codomain space W.

   Justification

   Definition of inner product space.

   Shown in the video
3. Expression

   $$\langle a, b \rangle_V = \langle F(a), F(b) \rangle_W$$
   Explanation

   Equate the two inner products to define the preservation property.

   Justification

   Condition for linear isometry.

   Shown in the video

The map F preserves the geometric structure measured by the inner product.

1. Audio
   Observation

   The speaker derives the property by explaining how coefficients are pulled out.

2. Formula
   Observation

   The speaker writes the final equation g(fX, hY) = fh $g(X,Y)$.

1. Expression

   $$g(fX, hY)$$
   Explanation

   Start with the tensor g evaluated on scaled vector fields fX and hY.

   Justification

   Definition of the inputs to the tensor field.

   Shown in the video
2. Expression

   $$= f h g(X,Y)$$
   Explanation

   Pull out the scalar functions f and h from their respective arguments.

   Justification

   Property of being a covariant tensor field (linear over functions).

   Shown in the video

A covariant 2-tensor field is linear over the ring of smooth functions, allowing coefficient functions to be factored out.

1. Audio
   Observation

   The lecturer says that because g restricted to each p is an inner product on the tangent space, for two vectors x and y one has $g(x,y)=g(y,x)$, so it is symmetric.

2. Formula
   Observation

   The board separately states that g is a symmetric 2-tensor field.

1. The spoken formula uses $g(x,y)$ without the subscript p, while the board's formal definition uses $g_p$ on T_pM; the intended pointwise reading is clear from context.

1. Expression

   $$g_p \text{ is an inner product on } T_pM$$
   Explanation

   Start from the defining property that the value of the metric at each point is an inner product on the tangent space.

   Justification

   Stated directly on the board in the definition of a Riemannian metric.

   Shown in the video
2. Expression

   $$g(x,y)=g(y,x)$$
   Explanation

   Because an inner product is symmetric, the lecturer writes the symmetry relation for two vectors x and y.

   Justification

   Spoken explanation in the audio.

   Shown in the video
3. Expression

   $$g \text{ is a symmetric 2-tensor field}$$
   Explanation

   Therefore the tensor field g is symmetric.

   Justification

   Conclusion stated both verbally and on the board.

   Shown in the video

The symmetry of g follows from the fact that each $g_p$ is an inner product on T_pM.

1. Formula
   Observation

   The board writes "positive definite in the sense that $g_p(v,v) \ge 0$, $g_p(v,v)=0 \Leftrightarrow v=0$".

2. Audio
   Observation

   The lecturer says it is also positive definite because at each point $g_p(v,v)$ is nonnegative and equals zero if and only if the vector itself is zero.

1. Expression

   $$g_p(v,v) \ge 0$$
   Explanation

   At each point p, evaluating the metric on a tangent vector against itself gives a nonnegative number.

   Justification

   Written on the board and repeated in the audio.

   Shown in the video
2. Expression

   $$g_p(v,v)=0 \Leftrightarrow v=0$$
   Explanation

   The quadratic expression vanishes exactly for the zero tangent vector.

   Justification

   Written on the board and explained verbally.

   Shown in the video

These two conditions express that $g_p$ is positive definite on T_pM.

1. Formula
   Observation

   The lower page shows "Rmk: Use a partition of unity.", then an atlas $\mathcal{A}$, a subordinate partition of unity \{This formula needs review. Please report it using the page feedback control.alpha\}, and "Now denote This formula needs review. Please report it using the page feedback control.alpha = $\sum_i$ du_\alpha^$i \otimes$ du_\alpha^i".

2. Audio
   Observation

   The lecturer says the next part will study existence of Riemannian metrics.

1. The clip ends before the construction using This formula needs review. Please report it using the page feedback control.alpha and This formula needs review. Please report it using the page feedback control.alpha is carried out to define a global metric.

2. The final assembly step of the proof is not visible in this segment.

1. Expression

   $$\text{Use a partition of unity.}$$
   Explanation

   The proof strategy is announced as using a partition of unity.

   Justification

   Written on the board under the proposition heading.

   Shown in the video
2. Expression

   $$\mathcal{A}=\{F_\alpha: U_\alpha \to Q_\alpha \subseteq \mathbb{R}^n\}$$
   Explanation

   Choose an atlas of the manifold M.

   Justification

   Written on the lower page.

   Shown in the video
3. Expression

   $$\{\rho_\alpha: U_\alpha \to [0,1]\}$$
   Explanation

   Take a partition of unity subordinate to that atlas.

   Justification

   Written on the lower page.

   Shown in the video
4. Expression

   $$\delta_\alpha = \sum_i du_\alpha^i \otimes du_\alpha^i$$
   Explanation

   On each chart, define a local covariant 2-tensor from the coordinate differentials.

   Justification

   Written on the lower page.

   Shown in the video

The clip shows only the initial setup of the standard partition-of-unity proof that every smooth manifold admits a Riemannian metric; the remaining construction is outside this segment.

1. Audio
   Observation

   The speaker gives an example of multiplying a vector field by a function.

2. Formula
   Observation

   The speaker writes 'fX'.

Illustrate how a vector field can be multiplied by a smooth function.

1. X is a vector field

2. f is a smooth function

Show the result of f times X.

1. Expression

   $$fX$$
   Explanation

   Multiply the vector field X by the smooth function f.

   Justification

   Definition of module structure over smooth functions.

   Shown in the video

fX is another vector field.

This is a standard construction in differential geometry where vector fields form a module over the ring of smooth functions.

1. Animation
   Observation

   The digital whiteboard scrolls through previous notes covering inner products, norms, polarization identity, orthogonal complement, orthonormal basis, and Gram-Schmidt algorithm.

1. Digital whiteboard notes

1. Scrolling down through sections 2.1 Definitions

1. Handwritten text format

Visual recap of prerequisite linear algebra concepts before introducing the new definition.

1. Animation
   Observation

   The speaker writes the equations for inner product preservation and vector space isomorphism properties next to the definition.

1. Equations

2. Purple ink

1. Appearance of $\langle a$, $b \rangle_V = \langle F(a)$, $F(b) \rangle_W$

2. Appearance of $F(a+b) = F(a) + F(b)$ and F(ka) = kF(a)

1. Definition text remains visible

Explicitly breaking down the verbal definition into formal mathematical equations.

1. Animation
   Observation

   The digital whiteboard scrolls down to reveal the definition of a Riemannian metric.

1. Digital whiteboard

2. Handwritten text

1. The view scrolls downwards.

1. The content of the whiteboard remains the same.

The scrolling action transitions the focus from the previous topic of linear isometry to the new topic of Riemannian metrics.

1. Animation
   Observation

   The speaker uses a purple pen to circle or underline key terms in the definition, such as 'inner product', 'smooth manifold', 'Riemannian metric g', 'smooth covariant 2-tensor field', and 'tensor'.

1. Handwritten text

2. Purple pen strokes

1. Specific words and phrases are circled or underlined.

1. The underlying text remains unchanged.

The highlighting emphasizes the crucial components of the definitions being discussed.

1. Animation
   Observation

   The video shows a digital whiteboard where the definition of a Riemannian metric is written and highlighted.

1. Handwritten text

2. Digital pen

1. The speaker writes additional symbols like $C^\infty(M)$, g, X, f, h, Y, and TM.

2. The speaker underlines and circles parts of the pre-written definition.

1. The core definition text remains on screen.

The visual progression follows the speaker's verbal explanation, breaking down the formal definition into its constituent parts and properties.

1. Animation
   Observation

   A purple circle is drawn around $g_p$ and then around T_pM on the definition line.

2. Audio
   Observation

   During this interval the lecturer is explaining that the restriction to each p is an inner product on the tangent space.

1. $g_p$

2. T_pM

3. purple annotation circle

1. First $g_p$ is circled.

2. Then T_pM is circled.

1. The underlying definition text remains unchanged.

2. The highlighted relation is that $g_p$ lives on T_pM.

The visual emphasis matches the verbal point that the metric is defined pointwise: at each p, $g_p$ is an inner product on the tangent space T_pM.

1. Animation
   Observation

   A purple underline is drawn beneath "positive definite" and then beneath the inequality line involving $g_p(v,v)$.

2. Audio
   Observation

   The lecturer says the metric is also positive definite and explains the nonnegativity and zero-equivalence condition.

1. the phrase "positive definite"

2. the formula $g_p(v,v) \ge 0$, $g_p(v,v)=0 \Leftrightarrow v=0$

3. purple underline

1. The phrase "positive definite" is underlined first.

2. The displayed inequalities are underlined next.

1. The definition text itself does not change.

2. The focus stays on the pointwise quadratic form $g_p(v,v)$.

The annotation directs attention to the exact algebraic meaning of positive definiteness in the definition.

1. Animation
   Observation

   The page scrolls downward from the definition block to the proposition heading and then further to the proof setup lines.

2. Formula
   Observation

   Newly revealed text includes "[Prop, Existence of Riemannian metric]", "Every smooth manifold admits a Riemannian metric.", "Rmk: Use a partition of unity.", the atlas definition, and This formula needs review. Please report it using the page feedback control.alpha = $\sum_i$ du_\alpha^$i \otimes$ du_\alpha^i.

1. definition paragraph

2. proposition heading

3. proof setup lines

1. The view moves from the definition of Riemannian manifold to the proposition statement.

2. It then reveals the first lines of the proof using an atlas and partition of unity.

1. The previously written definition remains the same above.

2. The mathematical topic stays on Riemannian metrics.

The scroll marks the transition from defining the central object to stating and beginning the proof of its existence on arbitrary smooth manifolds.

1. Audio
   Observation

   The speaker corrects himself from saying 'tensor field' to 'tangent spaces'.

Thinking that the vectors v and w mentioned in the definition belong to a tensor field directly.

The vectors v and w belong to the tangent space T_pM at a specific point p. The metric g itself is the tensor field that assigns an inner product to these tangent spaces.

1. Formula
   Observation

   The board defines g as a smooth covariant 2-tensor field whose value at each p is an inner product on T_pM.

2. Audio
   Observation

   The lecturer emphasizes the pointwise restriction to each p.

One might think of a metric as a single numerical function on the manifold rather than a tensor field varying from point to point.

The clip defines g as a smooth covariant 2-tensor field, and the key point is that at each point p the value $g_p$ is an inner product on the tangent space T_pM.

1. Formula
   Observation

   The board writes "$g_p(v,v) \ge 0$, $g_p(v,v)=0 \Leftrightarrow v=0$".

2. Audio
   Observation

   The lecturer explicitly says the equality to zero holds if and only if the vector itself is zero.

One may confuse positive semidefiniteness with positive definiteness.

In this definition, $g_p(v,v)$ is not only nonnegative; it vanishes exactly when $v=0$, which is the extra condition that makes the form positive definite.

1. Audio
   Observation

   Speaker mentions generalizing this notion to abstract manifolds using tangent spaces and differential mappings to study Riemannian metrics.

2. Formula
   Observation

   Section 2.2 Riemannian Metrics is visible on the board.

The concept of linear isometry between inner product spaces is generalized to define isometries between manifolds via their tangent spaces, which is essential for studying Riemannian metrics.

1. Audio
   Observation

   The speaker states, "We extend the above geometric notions to abstract smooth manifolds -> a smooth varying choice of inner product on each tangent space."

The concept of a Riemannian metric generalizes the idea of preserving inner products (as in a linear isometry) to smooth manifolds by assigning an inner product to each tangent space.

1. Formula
   Observation

   The definition states 'A Riemannian metric g ... is a smooth covariant 2-tensor field'.

The definition of a Riemannian metric contains the concept of it being a tensor field, which implies the linearity property discussed.

1. Formula
   Observation

   The speaker defines TM using the tangent spaces T_pM which are part of the manifold M.

The concept of a Riemannian metric applies to the tangent bundle, as it defines an inner product on each fiber (tangent space) of the bundle.

1. Formula
   Observation

   The board first defines a Riemannian metric g and then writes "Endowed with a Riemannian metric, (M,g) is called a Riemannian manifold."

The notion of a Riemannian manifold is built directly from the prior definition of a Riemannian metric on M.

1. Formula
   Observation

   After defining the metric and the manifold, the board introduces "[Prop, Existence of Riemannian metric]" and the statement that every smooth manifold admits one.

2. Audio
   Observation

   The lecturer says the next part will study existence of Riemannian metrics.

The existence proposition applies the just-defined notion of Riemannian metric to all smooth manifolds.

1. Audio
   Observation

   The lecturer derives symmetry of g from the fact that its restriction to each p is an inner product.

The derivation of symmetry depends on the defining requirement that $g_p$ is an inner product on T_pM.

1. Audio
   Observation

   Speaker defines linear isometry.

1. Definition of Linear Isometry

1. Formula
   Observation

   Equations for vector space isomorphism are written.

1. Conditions for Vector Space Isomorphism

1. Formula
   Observation

   The definition of a Riemannian metric is written on the board.

1. Riemannian Metric

1. Audio
   Observation

   The speaker explains, "covariant simply means it is a function of vectors. So you input some vectors and output is a number."

1. Riemannian Metric

1. Formula
   Observation

   The full definition is visible on screen.

1. Definition of Riemannian Metric
2. Positive Definiteness of Riemannian Metric
3. Symmetry of Riemannian Metric

1. Formula
   Observation

   The equation g(fX, hY) = fh $g(X,Y)$ is written.

1. Linearity Property of Tensor Fields
2. Derivation of Tensor Field Linearity

1. Formula
   Observation

   The equation TM = $\bigsqcup_{p \in M} T$_pM is written.

1. Definition of Tangent Bundle

1. Formula
   Observation

   The full definition of a Riemannian metric is written on the board.

1. Definition of a Riemannian metric

1. Audio
   Observation

   The lecturer explains symmetry via $g(x,y)=g(y,x)$ because each $g_p$ is an inner product.

1. Definition of a Riemannian metric
2. Why the metric is symmetric

1. Formula
   Observation

   The board displays $g_p(v,v) \ge 0$ and $g_p(v,v)=0 \Leftrightarrow v=0$.

1. Definition of a Riemannian metric
2. Pointwise positive definiteness of the metric

1. Formula
   Observation

   The board states that (M,g) endowed with a Riemannian metric is called a Riemannian manifold.

1. Definition of a Riemannian manifold

1. Formula
   Observation

   The lower page shows the proposition and the first lines of the proof using a partition of unity and This formula needs review. Please report it using the page feedback control.alpha.

1. Only the opening of the proof is present in this clip.

1. Existence proposition and proof setup
2. Existence of Riemannian metrics on smooth manifolds
3. Beginning of the existence proof via partitions of unity