# For diagonal metrics: Riemann curvature = 2-d Gauss curvature + ‘intermediary-dimension’ term

For diagonal metrics of 3 or more dimensions, the diagonal Riemann curvature tensor components Rlili are the sum of two pieces, the Gaussian curvature Kli for a 2-d surface, plus an ‘intermediary-curvature’ term reflecting the effect of the additional dimensions.

Writing ga for the positive sqrt of gaa : gaa= ea ga2 , the Gaussian curvature is: [No sum on A,B]:  (see Appendix 1)

KAB  =  – [gA gB]-1{h[gA,B /gB],+ hA[gB,A/gA],A}   .

The essential aspect is:   (1a)  where the 2-d curvature Kie is a sum of two such identical terms, just with i,e interchanged.

Note that the formula is identical under interchange of the two indices; indeed the two terms are identical under such interchange.

The below formula for the Riemann tensor component (RTC) formula for diagonal metrics, is from L. D. Landau, E. M. Lifshitz, Classical Theory of Fields (section 92)]; note that all the signature information has been separated out into the e’s:

gaa = ea exp(2Fa), where each F can be a different function, of various of the coordinates; (Font issues: the Summation is indicated via “SUM”).

Rlili = el e2Fl (Fi,iFl,i – F2l,i – Fl,i,i) +ei e2Fi(Fl,lFi,l – F2i,l – Fi,l,l)

– ei e2Fi el e2FlSUMm i,l eme-2Fm Fi,m Fl,m    i  (A)

(for the reader’s convenience, we have inserted the equation as a photo of the actual page in LL):

(i,l) are the coordinates of the 2-d ‘surface’, and m in the summation are the coordinates of what we will call the ‘intermediary dimensions’.

Note that the formula is identical under interchange i <–> l.

The first two sub-terms are identical under i <–> l.

Appendix 3 provides a direct calculation of the Riemann curvature, arriving after some effort at LL’s formula above.

What we wish to show: the two interchange-invariant terms of the Gaussian curvature are identical to those of the Riemann curvature formula. To show this we of course only have to show the identity of one from each pair; eg that (Fi,iFl,i – F2l,i – Fl,i,i) is basically the Gaussian curvature term [gA,B /gB],B.

Slightly adapted notation in rewriting LL’s formula (A):

1. we use h rather than e to reduce confusion with the e-as-exponential;

2. in the summed term we have moved ei e2Fi  out of the summation (‘SUM’):

Rlili = el e2Fl (Fi,iFl,i – F2l,i – Fl,i,i) + ei e2Fi(Fl,lFi,l – F2i,l – Fi,l,l)

– el ei em e2Fe2FSUM m i,l e-2Fm Fi,m Fl,m    i l      (A)

To make it precise we will need to raise two indices of Rlili ; we will be showing that:

Rlili = Kli  + additional term involving the other dimensional coordinates.

Adapting formula (A) to our notation: Since in many fonts i and l are very similar, we’ll use the letters (e,i) for coordinate indices rather than (l,i).

We can write the positive sqrt of gaa , symbolized by ga:

gaa= ea ga2 ,

Where ea is the signature.

Relating various notations:

gaa = ea exp(2Fa),  ga = exp(Fa),  Fa= ln ga  ;

(Fa),i = F’ = (ln ga)’ = ga,i/ga  .

The first term of Rlili in (A) above, rewritten using our notation, is:(2a)

Using this notation, we want to show that (2a) is the Gaussian curvature’s essential term (1a), ie that:

Outline of our procedure (done explicitly below): In (3) above, there are three subterms on the LHS and one on the RHS. Expanding the [ ],i terms on both sides gives 4 subterms on the LHS and two on the RHS. However, two of the four on the LHS exactly cancel each other, leaving two subterms. After raising two indices on the LHS the two subterms are identical to the two on the RHS.

Side-discussion:

• We can directly see that the Riemann curvature formula above has a 2nd derivative contribution only in the first term, and clearly that first term has no involvement of ‘the other dimensions’, only the two represented in the repeated subscript.

It is reasonable to expect that the Riemann curvature should somehow contain the Gaussian curvature of the 2-d aspect, and particularly that it should contain the Gaussian curvature’s 2nd derivative term. The other part of the Riemann curvature formula (the summation) has only 1st derivatives, and so there should be a 2nd derivative contribution in (2a) which at least contains within it the 2nd derivative part of (1a).

• Note the difference in the denominators of the term in (1a) and the term in (2a).                           For g = exp(F): In LL’s Riemann formula, the exponent function F of the metric coefficient appears; of course from calculus we know this is obtained by differentiating the exponential metric coefficient function and then dividing by the function itself, and indeed part of our task will be to show how the Riemann term with ge in the denominator can be transformed to have gi in the denominator.

Demonstrating the claim: (click this link or read the inserted page below)

Calculating the diagonal Riemann tensor components for diagonal metrics is therefore rendered quite simple: for each such component calculate the Gaussian curvature:

and then add the summation-term giving the contribution to the curvature of the interaction with the intermediary-dimensions.

See lecture utilizing this formula for calculation of the Riemann tensor components for the FRW & Schwarzschild metrics.

Speculations:

• Can it be that there is some invariant characterization of those spaces (curvatures) which allow for diagonal metrics?
• What is the equivalent in a non-metric connected space which allows for orthogonal coordinates?
• Are the Riemann curvatures of generally-curved metric spaces expressible as the curvature for a diagonal metric plus some other (perhaps intuitively-meaningful) term?
• Can the off-diagonal components for a diagonal metric also be expressed in a simple form? For them, LL gives:

DiscussionProperties of the Riemann curvature evident from the above formula (A): The signs of the three terms: For 4-d spacetime, whatever the choice of signature:

• If both (l,i) are spatial we can ignore el & ei. For this case there is a term in the summation m = t, and so if that term is non-zero then el ei em = – 1, and so the sign of the summation term will change.
• At most only one of (l,i) can be time, and if indeed one of them is, then there will be a relative negative sign between the first two subterms; but in that case m cannot be t and so there is no sign change in the last term.

Note that all second order derivatives originate with Fl,i,i and Fi,l,l , however these are 2nd derivatives of the exponential function, not of the metric coefficients.

We can see that the second order effect, the true Riemann curvature, arises from the metric coefficients of the two dimensions represented in the RTC indices (e,i), whereas the ‘intermediary’ dimensions (represented by m) gives rise only to products of first derivatives.

Also: (and this is true not only for diagonal metrics): since the second derivative is with respect to the ‘other’ coordinate, one only gets first derivatives of a metric function with respect to ‘its’ coordinate, for example the time-metric coefficient g00 will not have 2nd derivatives with respect to time, etc (see LL sec 95 re ‘peculiarities of the structure of the Einstein equaitons’). However, time-spatial Riemann components will have 2nd time derivatives with respect to the spatial coordinate. For example, for the FRW metric, Rtrtr has R..(t).

[1]

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Orthogonal coordinates, diagonal metrics

Notation: For an orthogonal metric of any dimensionality we can write: ds2 = gaadxadxa    (summed on a) . We define: g D  ≡ √|gDD|  ;  No summation over D: to indicate this in the following we will insert [NS])

We also define hand wD via:

wD =  gDdx D (No summation over D [NS: D])

ds2 ≡ ha (wa) 2  ≡ ha [ga]2 [dxa]2.  (summed over a: summation will be assumed in the below unless NS is specifically indicated) [The ha contain the signature information.]         (1)              [3][3]

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2-d and Gaussian Curvature

For a 2-d surface, using an orthogonal coordinate system (xA, xB), the  metric is:

ds2 = gAAdxA ² + gBBdxB ²    (no sum on A or B [NS: A,B]).

Gauss’s curvature formula for this surface is: [See Barrett O’Neill: “Semi Riemannian Geometry” p81]

KAB  =  – [gA gB]-1{hA [gA,B /gB],+ hB[gB,A/gA],A}    [NS: A,B].      (Formula C)

The Gaussian curvature arises from the interconnection of the two dimensions represented by the coordinates corresponding to the two indices of K (or of the metric coefficients corresponding to the coordinates). Quantitatively, the Gaussian curvature arises via – or is dependent – the derivatives of the metric coefficient corresponding to one coordinate, derived w/r/t the other coordinate. In this sense the Gaussian curvature is the curvature arising from a “mutual” interconnection. (Then a second derivative is formed, giving rise to second order terms.)

Intrinsic Curvature: The above formula also provides the intrinsic curvature of a 2-d space described by the above metric.

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Gauss’s Formula for the Curvature of a 2-d Surface, in his notation [4][1]

Given a 2-d surface with coordinate grid (u,v) on it, with the metric:

ds2 =  Edu2 + Gdv2

In terms of the quantities:

e = sqrt|E|, g = sqrt|G|      , e is the sign (my h)

and using the notation: gu = ¶ug    , (gu/e) v = ¶vug/e)

the curvature of the surface is given by Gauss’s formula:

Kuv = (-1/eg)[e1(gu/e) u + e2(ev/g)v]

Which is the same as our (formula C) above.

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Intrinsic Curvature for 3-d and higher

The formula for the Riemann Tensor Components (RTCs) for orthogonal metrics of any dimensionality is the Gauss term above plus an additional term which we call the ‘intermediary term’ I:

IAB =  hDhB (hA)  gD-2 [gA,D/gA][gB,D/gB] D A,B          (summed on D A,B)

=  [gA gB]-1    hDhB (hA)  gD-2 [gA,D][gB,D] D A,B

This term vanishes unless both of the metric coefficients corresponding to the indices of the RTC have non-vanishing derivatives w/r/t the same coordinate(s). We will term “intermediary coordinates” those coordinates w/r/t which both of the metric coefficients corresponding to the indices of the RTC have non-vanishing derivatives. We can then say that this term relates to the curvature arising from the interconnection of the two dimensions represented by A and B via the dimensions represented by the “intermediary coordinates”.

We can therefore write:     (Note: No Sum on A,B. However, D summed on DA,B)

RABAB  =  KAB  + IAB = – [gA gB]-1{ [gA,B /gB],+ hA hB[gB,A/gA],A + hDhB (hA)gD-2 [gA,D][gB,D] D A,B }

Where the first (Gaussian) part of the RTC is the curvature arising from the “mutual” interconnection, and the ‘additional’ (third) term is the curvature arising from the interconnection via the “intermediary” dimensions/coordinates.

• The terms are manifestly symmetric under A ßà B.  [(AR to AR: check the signature part of the last term)] Note that this is not true of the usual formulae for the RTC.
• This division of the RTC into “mutual” and “intermediary” terms grants this form of the RTC formula a certain intuitive basis.
• Use of this formula facilitates intuiting shortcuts in calculations of the RTC. (Some RTCs can be computed mostly ‘by inspection’ using this formula; the terms arising in the RTCs can be easily traced to the metric; some properties of special metrics are easily discernible eg for metrics with inverse relationship eg: gtt = -grr-1 [Rtt is a Laplacian]).

Analysis of the Formula : Note that when the two metric coefficients are functions of each other’s coordinate, ie:

gA = gA(xB),

gB = gB(xA),

then both terms can survive. If neither is a function of the other, both terms must vanish. [And of course if only one is a function of the other’s coordinate then only one term survives.]

These terms clearly involve a second derivative of each of the two metric coefficients involved, w.r.t. the other coordinate, ie gA,B,B  and gB,A,A . This makes sense as curvature since it takes into account a type of twisting of the metric basis. But note that the first derivative eg gA,B  is ‘normalized’ by dividing with gB before the second derivative is taken: eg the term is:

[gA,B /gB],B     ;

furthermore, the resulting term with the second derivative is normalized by the product of the two metric coefficients: eg:

-[gA gB]-1[gA,B /gB],B .

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Note: See my formula for the Ricci tensor components for an orthogonal metric: LL gives the formula:

Rii = Sl i (Fi,iFl,i – F2l,i – Fl,i,i + ei el e2(Fi – Fl) ) [Fl,lFi,l – F2i,l – Fi,l,l – Fi,lSm i,l Fm,l]

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Appendix:  Gauss & Riemann curvature formulae in terms of our (non-standard) “square-root connection coefficients”

For orthogonal coordinates:

GDaa = gaa,D/[-2gDD] = [haga²],D/[-2hDgD²] = 2hagaga,D/[-2hDgD²]

= -hDha [ga/gD][ga,D/gD]

We define what we can perhaps term ‘square-root connections’. [Note: the square root sign (with a ‘roof’ which is missing here), the radical, is a mnemonic symbol for this since it is somewhat similar to the Greek gamma.]:

Daa ≡ [ga,D/gD]

so that:

GDaa = -hDha [ga/gD]  √ Daa

….. …………

In this notation, Gauss’s curvature formula for a 2-d surface is:

KAB =  -[gA gB]-1{ BAA,B + hA hB ABB,A} = -[gA gB]-1{[gA,B /gB],B  + hA hB[gB,A/gA],A}

This measure provides a fully dependable measure of the curvature of the two dimensional surface: if it vanishes there’s actually no curvature, whatever the functions gA and gB are, however much they make the metric look as though it is that of a curved surface.

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Schematically, we can write

Gaussian curvature    K21 ~   G21,1   +  G12,2

symbolically:

RABAB  ~  KAB  +  AAD BBD    , D A,B  { ie D = C, E}

One can see the connection coefficients as expressing a measure of the interconnection of two dimensions, but a somewhat subjective or arbitrary measure, and the Gaussian curvature  – involving appropriate derivatives of connection coefficient. – gives a more sophisticated and also more objective measure of the curvature.

The Structure of the Riemann Curvature in terms of the Connection

CHANGE OF NOTATION: W’e’ll write both the radical and the gamma, but this does NOT mean the square root of the gamma!

f a 2-d ‘section’ were the totality of the space, the curvature would be: R1212 ~ G21,1  + G12,2

However since this 2-d ‘section’ is part of a 4-d space, both dimensions of the 2-d section, eg labeled by the coordinates 1 and 2, can interact with eg the dimension labeled by 3, so that there are non-vanishing connection curvature terms G3and G32  . If so, the product of these terms will appear in the curvature formula. Eg:

R1212 ~ G21,1  + G12,2  + G31G32

We would then refer to 3 as the ‘intermediary dimension/coordinate’.

Thus although the formulae for the curvature of 2-d surfaces in a 3-d flat space is the same as for the curvature of an intrinsically-curved 2-d space not embedded in any higher dimension., for a 2-d section in a 3 or higher-dimensional space such as the 4 dimensions of spacetime, we must take into account the indirect curvature deriving from the way that the various dimensions interact with each other, and this necessitates adding a term onto the curvature formula for curved 2-d surfaces.

The Riemann sectional curvature involves two dimensions principally, but also the additional dimensions. Structurally the Riemann curvature for the orthogonal metrics we’ll analyze in this book can be written as a sum of derivatives of certain G’s and products of other G’s:

R1212 ~ G21,1  + G12,2  + Ga1Ga2

where we allow a to run over all the ‘intermediary dimension/coordinates’.

In general the Riemann ‘sectional curvatures’ will be labeled by 4 coordinates, but for the orthogonal metrics we’ll analyze in this book two will repeat, and so the Riemann curvatures will be labeled by only two coordinates corresponding to two dimensions out of the 4, eg we’ll write RABAB , and the formula will involve terms relating to specific interconnections of only two dimensions within the totality, eg [gB,A/gA],A which relates to the interconnectedness of the dimensions coordinatized by A and B.

The Riemann Curvature Components (RCC) for orthogonal metrics of any dimensionality is the Gauss curvature for 2-d involving derivatives of connections, plus an additional term composed of a product of connections: eg the Riemann sectional curvature RABAB has the additional term:

The RTC will contain a product term for each coordinate D which is neither A nor B: in 4-d, there are therefore at most two such coordinates, D = C and D = E, and therefore only two ‘additional terms’.

Note that Gauss curvature involves connections of the type GABB whereas the connections in this ‘additional’ term are of other type, where the coordinate wrt which we take the derivative is also the coordinate of the metric term in the denominator: eg:

Gaba = ga,b/ga

Substituting A for a and D for b givesGAAD, and then substituting B for a and D for b gives GBBD . The full ‘additional’ term for the interconnection of the dimensions represented by the coordinates A and B is:

{hDhB (hA) gD-2 }GAAD GBBD  D A,B = {hDhB (hA) gD-2} [gA,D/gA][gB,D/gB] D A,B

where D is the ‘Dummmy’ summation index.

This additional term vanishes if one or both of the terms gA,D/gA and gB,D/gB vanish, ie if either one or both of gA,D and gB,D  vanishes. Thus the additional term vanishes unless both of the metric coefficients corresponding to the indices of the RTC (ie gAA and gBB) have non-vanishing derivatives wrt the same coordinate(s) D.

• We will term “intermediary coordinates” those coordinates D wrt which both of the metric coefficients gA and gB corresponding to the indices of the RTC have non-vanishing derivatives. We can then say that this term relates to the curvature arising from the interconnection of the (A,B) 2-section with the “intermediary dimensions” represented by the “intermediary coordinates” D.

We can call the new term the “Intermediary Term”: Both the A and B metric coefficients must have derivatives w.r.t. the “intermediary” coordinate D, otherwise this whole term vanishes

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Keep this structure of the Riemann curvature component in mind.

• The terms are manifestly symmetric under A ßà B.[(AR to AR: check the signature part of the last term)]
• The first (Gaussian) part of the RTC is dependent on (or quantify the curvature arising from) the interconnection of the two coordinates corresponding to the two indices of the RTC (or of the metric coefficients corresponding to the coordinates), specifically on the derivatives of the metric coefficient corresponding to one coordinate, derived w/r/t the other coordinate. In this sense it is the curvature arising from the “mutual” interconnection. The product term relates to the curvature arising from the interconnection of the (A,B) 2-section with the “intermediary dimensions” represented by the “intermediary coordinates” D.

• This division of the RTC into “mutual” and “intermediary” terms grants this form of the RTC formula a certain intuitive basis, and also helps intuit shortcuts in calculations of the RTC (some RTCs can be computed mostly ‘by inspection’ using this formula; the terms arising in the RTCs can be easily traced to the metric [some properties of special metrics are easily discernible eg for metrics with inverse relationship eg gtt = -grr-1 one obtains a Laplacian].

The Riemann Tensor Components (RTC) are composed of terms interrelating the metric coefficients of two dimensions at a time (eg one term is a measure of the curvature of the 2 dimensions represented by the coordinates A and B, another is a measure of the curvature of the 2 dimensions represented by the coordinates by A and D, another for B and D). In general one must compute the Riemann components for all pairs of two indices, that is, one computes the curvatures in each “2-section” (A,B) as follows:

RABAB = -[gA gB]-1{[gA,B /gB],B + hA hB[gB,A/gA],A}hDhB  gD-2 [gA,D/gA][gB,D/gB]} DA,B            [5][4]

The Riemann tensor as a whole takes into account the curving of the whole spacetime by computing the curvature of each individual 2-surface of the whole spacetime. The field equation will have on the left hand side a combination of these individual curvatures, adding them together and making the sum equal to the tensor element on the right hand side, which is some aspect of the source (an element of the source tensor).

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Introduction: Define the basis one-form wA and the connection two-form wAD.

wA = gA dA

wAD clearly involves a first derivative of the one of the two metric coefficients involved, w.r.t. the other coordinate…..

dwAB  involves a second derivative of each of the two metric coefficients involved, w.r.t. the other coordinate. Therefore the terms in the RTC coming from dwAB  will involve a second derivative of each of the two metric coefficients involved, w.r.t. the other coordinate. This makes sense as curvature since it takes into account a type of twisting of the metric basis.

The RTC can be read from the coefficients in the “curvature two-form” RAB :

RAB = dwAB – wAD^ wDB  : summed over DA,B                                                                            (1)

The origin of the term in the RTC involving the product of the connection two-forms can be seen via construction of the RTC via parallel transport around a closed circuit, or via the two-form method, there will also be a In general the Riemann Tensor Components (RTC) for an orthogonal metric are composed of terms coming from one or both of dwAB  and  wAD ^ wDB   D A,B  ; for some special symmetry situations only one of these terms survives, whereas for non-orthogonal coordinates there are more than these two terms.

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The RTC are given by: [I have to check the signs]:

RABAB = -[gA gB]-1{[gA,B /gB],B + hA hB[gB,A/gA],A}hDhB  gD-2 [gA,D/gA][gB,D/gB] DA,B          (2)

Defining:

gA,B /gBBAA

the relationship to the connection coefficients (Christoffel symbols) is:

GBAA = ½ [gAA,B /gBB] =  ½ [gA2,B /gB2] = ½  2 gAgA,B /gB2 = [gA/gB ]  gA,B /gB  = [gA/gB ] BAA ;

GBBD = ½ [gBB,D /gBB] = ½ [gB2,D /gB2] = ½ [2gBgB,D /gB2] = [gB,D /gB] ≡BBD

See file: “Comparing my RTC formula to the usual one

Note:

v     No matter how many dimensions the space each term in the RTC above is computed only in a 2-d subspace, either (A,B), or (A,D), or (B,D)  ; this makes each such calculation relatively simple.

[ADVANCED: This is so since each connection two-form wAD depends only on the two dimensional subspace (A,D), and the “curvature two-form” RAB is composed of these connection two-forms]

v     The first two terms of the RTC above are simply the Gaussian curvature for the 2-d surface (A,B):

KAB = -[gA gB]-1{[gA,B /gB],B + hA hB[gB,A/gA],A}              (3)                       [6][5]

[ADVANCED The whole term [gA,D/gA][gB,D/gB] vanishes unless both the A and B metric coefficients have derivatives w.r.t. the “intermediary” coordinate D. From now on, this is the sense in which we will use the term “intermediary” coordinate.

v       [ADVANCED wAD ^ wDB   D A,B  contributes a term to RABAB proportional to: [gA,D/gA][gB,D/gB]  (summed over D)    [7][6]

Curvature of a 2-d Surface: Subspace of a Higher-d Space

For a 2-d space, ie a surface, there are only two orthogonal coordinates A and B, and therefore there is no D A,B  coordinate to be summed over, and so there can be no “intermediary” term [ADVANCED:wAD ^ wDB].  Therefore the RTC for a 2-d space (A,B) [ADVANCED: depend only on the terms arising from dwAB , namely]

RABAB = -[gA gB]-1{[gA,B /gB],B + hA hB[gB,A/gA],A}

which is in fact Gauss’s formula for 2-d surfaces.

In a higher-d space containing this 2-d subspace (A,B), the RTC will then simply be the Gaussian curvature for the 2-d surface with the addition of the intermediary term:

RABAB = KAB + [gA,D/gA][gB,D/gB] (summed over D)                                    (3)

v     We will term “self-contained” those subspaces in which the intermediary term vanishes (ie one or both of the A and B metric coefficients have no derivatives w.r.t. the “intermediary” coordinate D). Thus, “self-contained” 2-d subspaces have the same curvature as they would if there were no other dimensions: ie their RTC is equal to their Gaussian curvature.

v     The term in the RTC coming from the dwAB is:

[gA,B /gB],B + hA hB[gB,A/gA],A               (4)

When in the metric of the 2-d surface (A,B) the metric coefficients are functions of each other’s coordinate, then both terms survive. If neither is a function of the other, both terms vanish. [And of course if only one is a function of the other’s coordinate then only one term survives.]

[In general the RTC are composed of terms coming from both dwAB  and  wAD ^ wDB   D A,B  .]

v     But note that the first derivative is ‘normalized’ before the second derivative is taken: eg  [gA,B /gB],B  , and the resulting second derivative is normalized by the product of the two metric coefficients: -[gAgB]-1{[gA,B /gB],B .

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Appropriateness of the Word ‘Curvature’ as Applied to Spacetime Warping

The Gaussian curvature measure applies to an ordinary 2-dimensional surface in ordinary three-dimensional space, and we are very easily intuitively aware of the meaning of the word ‘curvature’ in that context. Although the meaning of the word ‘curvature’ is in a sense allegorical when applied to higher dimensions since we can never observe it directly as we do for 2-surfaces, and is certainly allegorical when applied to a combination of space and time as for spacetime, nevertheless since the Riemann ‘curvature’ is so closely modeled on the 2-d Gaussian curvature, we have good reason to maintain usage of the word ‘curvature’ even when applied to higher-dimensional spacetime warping.

Photos of some of the pages 9earlier verison), included for possible reader convenience

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See lecture utilizing this formula for calculation of the Riemann tensor components for the FRW & Schwarzschild metrics.