GEOMETRIC INTERPRETATIONS OF CURVATURE. Contents 1. Notation and Summation Conventions 1

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1 GEOMETRIC INTERPRETATIONS OF CURVATURE ZHENGQU WAN Abstract. Ths s an exostory aer on geometrc meanng of varous knds of curvature on a Remann manfold. Contents 1. Notaton and Summaton Conventons 1 2. Affne Connectons 1 3. Parallel Transort 3 4. Geodescs and the Exonental Ma 4 5. Remannan Curvature Tensor 5 6. Taylor Exanson of the Metrc n Normal Coordnates and the Geometrc Interretaton of Rcc and Scalar Curvature 9 Acknowledgments 13 References Notaton and Summaton Conventons We assume knowledge of the basc theory of smooth manfolds, vector felds and tensors. We wll assume all manfolds are smooth,.e. C, second countable and Hausdorff. All functons, curves and vector felds wll also be smooth unless otherwse stated. Ensten summaton conventon wll be adoted n ths aer. In some cases, the ndex tyes on ether sde of an equaton wll not match and so a summaton wll be needed. The tangent vector feld x nduced by local coordnates x wll be denoted as. 2. Affne Connectons Remann curvature s a measure of the noncommutatvty of arallel transortaton of tangent vectors. To defne arallel transort, we need the noton of affne connectons. Defnton 2.1. Let M be an n-dmensonal manfold. An affne connecton, or connecton, s a ma : XM XM XM, where XM denotes the sace of smooth vector felds, such that for vector felds V 1, V 2, V, W 1, W 2 XM and functon f : M R, 1 fv 1 + V 2, W = f V 1, W + V 2, W, 2 V, aw 1 + W 2 = a V, W 1 + V, W 2, for all a R. 3 V, fw = V fw + f V, W. We wrte V, W = V W. 1

2 2 ZHENGQU WAN Theorem 2.2. For fxed W, V W only deends on V. Proof. It suffces to show f V = 0 then U W = 0. The roof reles on the followng lemma: f V = 0 then there exsts smooth scalar felds f k and vector felds Ṽk such that f k = 0 and V = k f kṽk. Ths fact s easly roven usng arttons of unty. Usng ths lemma, we then wrte V = k f kṽk where f k = 0. Then k f W = f kṽk k Ṽk W = 0. k Remark 2.3. Snce V W deends on V ontwse, we may wrte v W := V W where v = V T M. We also note that V W deends on the local values of W, that s, f W 1 = W 2 n some neghborhood of, then v W 1 = v W 2 for all v T M. We now gve an exresson for the connecton n local coordnates x on an oen set U. Suose we have two vector felds W = w j j and V = v on U. Then W V = w j v x j + w j v j. We defne functons Γ k j : U R, for 1, j, k n, by j = Γ k j k. These functons are called the Chrstoffel symbols and deend on the coordnate system used to defne them. Snce covarant dfferentaton takes values n XM, we see mmedately that the Chrstoffel symbols are smooth. We now show exstence of a global affne connecton on a manfold M. Theorem 2.4. For any manfold M, there exsts an affne connecton on M. Proof. Let {U α } α be an atlas of M and let {ρ α } α be the assocated artton of unty. On each oen set U α, we have a connecton α gven n the coordnates on U α by Γ k j = 0. We defne by V W = α α ρα V ρ α W. Note that at each ont, s a fnte sum. Some straghtforward comutatons wll show that satsfes the three roertes n Theorem 2.1 and V W s smooth when V, W are smooth. Remark 2.5. The affne connecton on a manfold need not be unque. If we vary the Chrstoffel symbols Γ k j n each of the charts U α of the roof, we may get dfferent affne connectons. Remark 2.6. We know that w V deends on W locally, however we can further show that f η s a smooth curve such that η0 = and η 0 = w, then the value of w V only deends on behavor of V on the curve η. Thus, w V s well-defned even f V s only defned along a curve through and tangent to w. Defnton 2.7. Let η : I M be a smooth curve. A vector feld V along η s a smooth ma V : I T M such that for each t I, V t T ηt M. Remark 2.6 above says that for a vector v, the ma W v W s defned on vector felds W along curves η such that η0 = and η 0 = w. Defnton 2.8. A vector feld V along a curve η s arallel f ηt V = 0 for all t.

3 GEOMETRIC INTERPRETATIONS OF CURVATURE 3 Theorem 2.9. Let η : [0, 1] M be a smooth curve and let v T η0 M. Then, there exsts a unque arallel vector feld V along η such that V 0 = v. Proof. Frst, we consder the case when η s contaned n a local coordnate chart. In local coordnates, V t = V k t k satsfes the system of ODEs dv k dt + V η j Γ k j = 0 for k = 1,..., n wth ntal condtons V k 0 = v k. Observe that the ma V k, t Γ k jηtv j η j t s contnuous; for fxed t, the ma V k Γ k j V j η j s lnear and thus globally Lchtz contnuous n V k. Pcard-Lndelof theorem tells us that ths equaton has a unque soluton V t. For general η, we let 0 = t 0 < < t N = 1 be a artton of [0, 1] such that, for each, η [t 1,t ] s contaned n a coordnate chart U. The revous aragrah mles that there s a arallel vector feld V along η [0,t1] such that V 0 = v. If we have a arallel vector feld V along η [0,t], we obtan a vector feld Ṽ along η [t,t +1] such that Ṽ t = V t, agan usng the revous aragrah. Then, V extends to a vector feld along η [0,t+1] by defnng V t = Ṽ t for t [t, t +1 ]. By nducton, we get a vector feld V whch s arallel along all of η. 3. Parallel Transort One roblem on general manfolds s, unlke Eucldean saces, f, q M are dstnct onts, then there s no natural dentfcaton T M = T q M. In ths secton, we llustrate how we may construct such an somorhsm, called arallel transortaton, usng the noton of arallel vector felds. However, ths somorhsm wll deend on a choce of ath between and q. Defnton 3.1. Let η : [0, 1] M be a curve connectng two onts, q M,.e. η0 = and η1 = q. Let v T M and let V denote the arallel vector feld along η ensured by Theorem 2.9. The vector V 1 s called the arallel transort of v along η and the ma v V 1 s denoted T η0 η1 or T q or by T f the ath η s clear from the context. Remark 3.2. Snce the equaton defnng arallel transortaton s a lnear ODE, for any ath η from to q, we have T q av + w = at q v + T q w. From the defnton of arallel transort we see that T q s the nverse of T q. Thus T q s an somorhsm from T M to T q M. Remark 3.3. Let η be a curve and = η0. If {v 1,..., v n } s a bass of T M, then we can extend v to arallel vector felds E t along η. Snce arallel transortaton s nvertble, {E 1 t,..., E n t} forms a bass of T ηt M at each t. We have defned arallel transort n terms of the affne connecton. The followng theorem shows that we may defne the affne connecton n terms of arallel transort. Theorem 3.4. Suose we have a smooth vector feld V and a smooth curve η such that η0 = v. Then lm ɛ ɛ Tηɛ η0 V ɛ V 0 = η0 V.

4 4 ZHENGQU WAN Proof. Let E 1,..., E n be arallel vector felds along η formng a bass at each ont. Wrte V t = V k te k t. Then, T ηɛ η0 V ɛ = V k ɛe k 0, snce T ηɛ η0 s a lnear ma. Then, 1 ɛ T ηɛ η0v ɛ V 0 = 1 ɛ V k ɛ V k 0E k 0. Takng the lmt ɛ 0 gves us d dt t=0v k te k 0. Snce E k s arallel, we also have that η 0V t t=0 = η 0V k te k 0 + V k 0 η 0E k t = η 0V k te k 0. Ths s, by defnton d dt t=0v k te k 0, so the two quanttes are the same. Remark 3.5. Note that ths theorem states that the affne connecton s equal to to the lmt of a dfference quotent. For ths reason, for a fxed curve η, the ma V t ηt V t, takng a vector feld V along η to a vector feld along η, s sometmes called the covarant dervatve along η. Defnton 3.6. Let be an affne connecton on M. Gven a vector v and an tensor feld T, we defne the tensor v T of the same rank as T nductvely by the followng roertes 1 f f : M R s a smooth functon,.e. a 0, 0-tensor, then v f = vf, 2 f W s any vector feld, then v W = v W, 3 v T s R-lnear n v and T, 4 Lebnz v T S = v T S + T v S, 5 If Contract b at denotes contractng T wth resect to ndces a and b. Then v Contract b at = Contract b a v T. Note that roertes 1 and 2 ndcate that extends the connecton to all tensor felds. Ths defnton wll be of nterest n order to cut down on notaton. Theorem 3.7. Gven coordnates x, we have j dx = Γ kj dxk. Proof. Let j dx = kj dxk, where {dx } s the bass of T M dual to { j }. Notce dx l = δ l s a constant functon, so Thus, l j = Γl j. 0 = j [dx l ] = j dx l + dx l j = Γ l j + l j. 4. Geodescs and the Exonental Ma Geodescs are the analogue of straght lnes n Eucldean sace and ossess many of the same roertes as straght lnes. Defnton 4.1. A smooth curve η : a, b M s called a geodesc f the tangent vector feld η along η s arallel,.e. f ηt ηt = 0 for all t a, b. Theorem 4.2. Gven a ont M and a vector v T M, there exsts an ɛ > 0 and a geodesc η : ɛ, ɛ M such that η0 = and η 0 = v. Proof. In terms of local coordnates, the geodesc equaton ηt ηt = 0 becomes d 2 η k dt 2 + dη dη j Γk j = 0 dt dt along wth the ntal values η0 = and η 0 = v. The theorem follows by exstence and unqueness of solutons to systems of ODEs.

5 GEOMETRIC INTERPRETATIONS OF CURVATURE 5 Note that the theorem does not ensure unqueness of geodescs, snce two geodescs may not be defned on the same nterval around 0. We defne a maxmal geodesc to be a geodesc γ : I M, where I s some nterval, wth the roerty that for any extenson of γ to a new geodesc η : J M, we have I = J and η = γ. Defnton 4.3. Gven M and v T M, there s a unque maxmal geodesc η assng through wth velocty v. The exonental ma at, Ex : T M M, s defned by v η1. Note that ths ma s only defned on a subset of T M. Theorem 4.4. There s an oen neghborhood U around 0 T M such that Ex U s a dffeomorhsm onto ts mage. Remark 4.5. The exonental ma s a local dffeomorhsm, and so we can arametrze our manfold M near by neghborhood of orgn n T M as follows. Let U be a neghborhood of n M and Ũ a neghborhood of 0 n T M such that Ex Ũ s a dffeomorhsm onto U. If {v 1,..., v 2 } s an orthonormal bass for T M, then we can defne a ma takng q U to the comonents of Ex Ũ 1 q n the bass {v }. Ths coordnate chart s called the normal coordnate chart at assocated to the bass {v k }. It s a useful comutatonal tool that wll be emloyed frequently. 5. Remannan Curvature Tensor Parallel transortaton s ath-deendent; gven two aths η 1 and η 2 wth the same endonts, arallel transortaton along η 1 and η 2 are, n general, not the same. Remann curvature measures how much arallel transort deends on the ath. Defnton 5.1. Gven an affne connecton and two vector felds X, Y, we can defne another vector feld T X, Y = X Y Y X [X, Y ]. The ma T s called the torson tensor. Theorem 5.2. The torson tensor s a tensor feld. Proof. We just have to show that T s C -lnear n ts arguments. T fx, Y = fx Y Y fx [fx, Y ] = f X Y f Y X + Y fx f[x, Y ] Y fx = f X Y Y X f[x, Y ] = ft X, Y Defnton 5.3. An affne connecton s called torson-free f T = 0. Remark 5.4. In local coordnates, the torson-tensor T has an exresson gven as follows. Let X = X and Y = Y j j, X Y Y X [X, Y ] = Γ kj Γ kjx Y j k. Notce that the exresson only nvolves the comonents of X and Y and not ther dervatves. Thus T = 0 s equvalent to Γ k j = Γk j,.e. the Chrstoffel symbol s symmetrc n ts lower two ndces. On a Remannan manfold M, g, there s a unque affne connecton satsfyng two addtonal roertes, torson-freeness and comatblty wth the metrc.

6 6 ZHENGQU WAN Theorem 5.5 Fundamental Theorem of Remannan Geometry. Let M, g be a Remannan manfold. Then there exsts a unque torson-free affne connecton on M such that v g = 0 for all tangent vectors v. The second roerty v g = 0 s equvalent, by defnton, to vgy, Z = g v Y, Z + gy, v Z for any vector felds Y, Z and any tangent vector v. Ths s also equvalent to requrng for all arallel vector felds V t, W t along a curve η, ther nner roduct s reserved,.e. for all t, gv t, W t = gv 0, W 0. Proof. To rove unqueness, let be a torson-free affne connecton such that v g = 0 for all tangent vectors v. We frst comute g jk = g j, k = g j, k + g j, k = g lk Γ l j + g jl Γ l k. Notce g j s symmetrc n and j and Γ k j s symmetrc n and j by torsonfreeness. By ermutng the ndces, j, k, we obtan the followng three equatons. g kl Γ l j + g jl Γ l k = g jk g jl Γ l k + g l Γ l jk = k g j g l Γ l kj + g kl Γ l j = j g k. If we add the frst two equatons and subtract the last we fnd Therefore g jk + j g k k g j = 2g kl Γ l j. Γ l j = 1 2 gkl g jk + j g k k g j. Ths shows that s unque. For exstence, we substtute the coordnate exresson for Γ k j nto the roof of Theorem 2.4. It s then easy to check that the resultng affne connecton s torson-free and v g = 0 for all tangent vectors v. We thus have the followng defntons. Defnton 5.6. The Lev-Cvta connecton on a Remannan manfold M, g s the affne connecton satsfyng the addtonal roertes: 1 s torson-free, 2 w g = 0 for all tangent vectors w. Defnton 5.7. Gven vector felds U, V, W, we defne the Remann curvature R by RU, V W = U V W V U W [U,V ] W, where s the Lev-Cvta connecton. Theorem 5.8. The Remann curvature R s a tensor feld.

7 GEOMETRIC INTERPRETATIONS OF CURVATURE 7 Proof. It suffces to show that R s C M-lnear n each argument. C M, then U V fw = UV f + V f U W + Uf V W + f U V W V U fw = V Uf + Uf V W + V f U W + f V U W [U,V ] fw = [U, V ]fw + f [U,V ] W. Substtutng these values nto the defnton of Remann curvature gves RU, V fw = f U V W f V U W f [U,V ] W = fru, V W If f Thus R s C M-lnear n the thrd varable W. Smlar comutatons show that R s also C M-lnear n frst two varables. Remark 5.9. Snce R s a tensor feld, RU, V W deends only on the values of U, V, and W at the ont. In terms of local coordnates, f U = U, V = V j j, and W = W k k, then RU, V W = R l jk U V j W k. We have R l jk = dx l R, j k = dx l j k j k [, j] k = Γl kj x Γl k x j + Γα kjγ l α Γ α kγ l αj Theorem The Remann curvature tensor satsfes, for all tangent vectors U, V, W, X, Y based at the same ont, 1 RV,U=-RU,V 2 Banch dentty RU,VW+RV,WU+RW,UV=0 3 gru,vx,y=-gx,ru,vy Proof. The frst equalty follows drectly from the defnton of R. The last follows by substtutng gx, Y for W n the exresson for R: 0 = UV gx, Y V UgX, Y [U, V ]gx, Y and notng that ths s zero and alyng the Lebnz rule roerty 2 n Defnton 5.6. Ths s equal to 0 = Ug V X, Y + UgX, V Y V g U X, Y V gx, U Y g [U,V ] X, Y gx, [U,V ] Y = gru, V X, Y + gx, RU, V Y. The last second dentty s checked by drect comutaton and s omtted. The defnton of Remann curvature has been extremely abstract. We now see one of t s very nce geometrc meanng. Suose we have two lnearly ndeendent vectors u, v T M. Then we can fnd local coordnates x n whch 1 = u and 2 = v. Consder the four vertces {0, 0, s, 0, s, r, 0, r} of a rectangle n the x 1 x 2 -lane, llustrated by the followng dagram. We denote ts edges by A, B, C, D, startng from 0, 0 n the counterclockwse drecton. Let w be a tangent vector at 0, 0. We let T denote arallel transortaton around the loo ABCD. Note that ths oeraton s deendent on s and t. Theorem lm s,t 0 w T w st = Ru, vw

8 8 ZHENGQU WAN Proof. By abuse of notaton, A, B, C, D wll not only denote the aths but also the oeraton of arallel transortaton along these aths,.e. A wll denote arallel transortaton along the ath A, B wll denote arallel transortaton along B, etc. Thus, T = DCBA. We may assume that the aths A, B, C, and D are contaned n a sngle coordnate chart. Extend u, v, w to vector felds U, V, W near 0, 0. We may assume that U and V are coordnate vector felds: U = s and V = r such that [U, V ] = 0. Notce W T W = DCC 1 D 1 W BAW and C 1 D 1 W BAW We observe = C 1 D 1 W W + C 1 W W BAW W BW W C 1 D 1 W W lm r 0 r lm r 0 BW W r Then C 1 D 1 W W BW W lm s,r 0 sr We also have C 1 W W lm s 0 s lm s 0 BAW W s = C 1 V W = V W = lm s 0 C 1 V W + V W s = U W = B U W = U V W Then C 1 W W BAW W U W + B U W lm = lm = V U W s,r 0 sr r 0 r Therefore f we combne the above exressons we have W T W lm s,r 0 sr = lm s,r 0 DC U V W V U W = U V W V U W = RU, V W Remark If s the Lev-Cvta connecton of a Remann manfold, then RU, V s a skew-symmetrc oerator,.e. for all X and Y, g RU, V X, Y = g X, RU, V Y. Ths was roved n Theorem 5.10, 2. However, ths can also be seen because arallel transortaton T reserves the nner roduct g, therefore s an element of the orthogonal grou OT M = On. The oerator RU, V s the dervatve of ths arallel transortaton and the tangent sace T Id On of the sace orthogonal transformatons On s the sace of skew-symmetrc transformatons. We now gve two addtonal tensor felds whch are also called curvature. Defnton We defne the followng:

9 GEOMETRIC INTERPRETATIONS OF CURVATURE 9 1 The Rcc curvature Rc s gven by secfyng a local orthonormal frame {E } n =1 and comutng n RcX, Y = grx, E Y, E. =1 Ths does not deend on the choce of E, 2 The scalar curvature S s the trace of the Rcc curvature,.e. for an orthonormal frame E, we have S = n =1 RcE, E. Locally, the Rcc curvature s gven by the contracton Rc j = Rkj k and scalar curvature s the contracton S = Rc = R j j. 6. Taylor Exanson of the Metrc n Normal Coordnates and the Geometrc Interretaton of Rcc and Scalar Curvature In ths secton, we wll see how Rcc curvature measures the devaton of the Remannan metrc from the standard Eucldean metrc and how scalar curvature measures the devaton n the volume of a geodesc ball from the volume of a Eucldean ball of the same radus. Choose an orthonomal bass {e k } k of T M and a neghborhood U of, on whch the exonental ma Ex s a dffeomorhsm. Let U, x k denote the normal coordnates at assocated to the bass {e k }. In ths coordnate chart, we have 1 g j = δ j 2 Γ k j = 0 3 gj x k = 0 We rove 1 by g j =, j = e, e j = δ j We rove 2 by usng the fact that for each v T M, ηt = Ex tv s a geodesc and thus satsfes the geodesc equaton Therefore we have d 2 η k dt 2 + dη dη j Γk j dt dt 0 + Γ k jv v j = 0 where v = v k e k. We restrct the equaton to the ont and choose an arorate value of v to get Γ k j + Γ k j = 0, for all, j, k. = 0 Snce the Lev-Cvta connecton s torson-free, we conclude Γ k j = 0 Equaton 3 follows by comutng g j x k = k e, e j = k e, e j + e, k e j = Γ l kg lj + Γ l kjg l Snce the Chrstoffel symbol vanshes at, we conclude the dervatve of g vanshes n normal coordnates.

10 10 ZHENGQU WAN Theorem 6.1 Gauss Lemma. Let x be normal coordnates around a ont. Then g satsfes x k q = g kj q x j q, j for all onts q n the coordnate neghborhood and for all k. Before we gve the roof, we need a short lemma. Lemma 6.2. Let q be a ont n a normal coordnate chart at, wth coordnates x q = x. Then x 2 = g j q x x j. Proof. The curve η t = x t n the normal coordnate s a geodesc on a Remannan manfold. Therefore, g η0, η0 = g η1, η1 whch shows that δ j x x j = g j q x x j. Proof of Gauss Lemma. Frst observe that f the coordnate system x s normal then the straght ath η k t = x k t = x 1 t,..., x n t s a geodesc through. If ηt = x k t s a geodesc, substtutng ηt nto the geodesc equaton we get Γ k jηtx tx j t = 0. We substtute n the formula for Γ k j and snce gj s nvertble, we get 1 gk 2 x j + g jk x g j x k x x j = 0. For convenence, we dro the argument showng that the exresson on the left s evaluated along ηt. We wll contnue to do ths, keeng n mnd that all exressons are to be evaluated along ths curve. We note that g k x x x j = g jk j x x x j, and so gk 6.3 x j 1 g j 2 x k x x j = 0. We ntroduce the functon 6.4 x β = α g αβ x α and we wsh to show that x β = x β for each β. Dfferentatng wth resect to x δ gves x β x δ = g βα x δ xα + g βδ α We substtute 6.3 nto ths equalty to obtan, after some mnor algebrac manulaton, = j = j x k x j g kj x j 1 2 x k x j xj 1 x x 2 x k x x k g k Now by Lemma 6.2 and Equaton 6.4 we have x x = g j x x j = x 2,j x

11 GEOMETRIC INTERPRETATIONS OF CURVATURE 11 and so substtutng ths nto Equaton 6.5 gves x k x j xj x k = x k x k x j x j = 0. j j Snce ths holds for all geodescs ηt, we get that x k x k x j = 0. At, we have x k = x k, therefore we have roven that at any ont n the normal coordnate we have x k = x k = j g kjx j. The dervatves of the Gauss Lemma can tell us useful nformaton about the dervatves of the metrc. Startng from dfferentate wth resect to : x k = j g kj x j, δ k = g kj x xj + g k. Evaluatng at the orgn we get Dfferentatng agan wth resect to l, Evaluatng at the orgn we get gkj Reeatng the rocess we get 2 g km x x l + g k = δ k. 0 = 2 g kj x x l xj + g kl x x + g k x l. + g k x j = 0. 2 g k x l x m + 2 g kl x x m = 0. A consequence of ths constrant on the second dervatves of g s that g j 2 x l x k x k x l = 1 3 R kjl x k x l. See Secton 3 and 4 of [1] for detals of dervaton of 6.6. Therefore, we have the Taylor exanson of the Remannan metrc g n local coordnates as g j = g j + g j x k x k g j 2 x k x l x k x l + O x 3 = δ j 1 3 R kjl x k x l + O x 3 Defnton 6.7. The Remann volume element s the unque volume element ω on a orented Remann manfold such that for any ostvely orented orthonomal bass {v 1,..., v n } n a tangent sace we have ωv 1,..., v n = 1.

12 12 ZHENGQU WAN Remark 6.8. In local coordnates x, ω = Detg j dx 1 dx n. We now study the Taylor exanson of the Remann volume element. Recall the formula DetId + A = 1 + TrA + O A 2, where A = max{ a j } s the maxmum absolute value of all entres of A. We have 1 Detg j = 1 Tr 3 R kjlx k x l + O x 3. Therefore, 1 Detg j = 1 Tr 3 R kjl x k x l + O x 3 = Rc kl x k x l + O x 3 = Rcx, x + O x 3 Where Rc s the Rcc curvature. Because g j = δ j, rasng and lowerng ndces does not have an effect on the comonent of the curvature tensor. Fnally, usng the Taylor exanson of the square root functon, Theorem 6.9. In normal coordnates x around, we have the local exresson ω = Detg j dx 1 dx n = Rc x, x + O x 3 dx 1 dx n. Therefore we observe a nce geometrc meanng of the Rcc curvature. It tells us how the Remann volume element ω devates from the standard Eucldean volume element ω E = dx 1 dx n n the normal coordnates. Defnton A geodesc ball B, r s collecton of onts x = x n the normal coordnate chart at such that x 2 r 2. Assumng r s small enough such that the exonental ma s njectve on the ball We nvestgate the volume of the geodesc ball B, r. We let B E 0, r denote the standard Eucldean ball of radus r and V B the volume of a ball B and by V B the volume of ts boundary B. V B, r = ω = B,r x B E 0,r Rc x, x + O x 3 dx 1 dx n. Snce Rcc curvature s a symmetrc blnear form, t has orthonormal rncal axs {v 1,..., v n } and corresondng egenvalues {λ 1,..., λ n }. Let y be the

13 GEOMETRIC INTERPRETATIONS OF CURVATURE 13 normal coordnates corresondng to the bass v. Therefore, Rcx, x dx 1 dx n x B E 0,r = y B E 0,r = λ = λ λ 1 y λ n y n 2 dy 1 dy n y B E 0,r 1 n y 2 dy 1 dy n ρ 2 dy 1 dy n, y B E 0,r where ρ 2 = y 2 and the sum λ = S, the scalar curvature at the ont. The ntegral can be evaluated usng hyershercal coordnates n n-dmensons: ρ 2 dy 1 dy n = ρ 2 ρ n 1 dρ dω. y B E 0,r B E 0,r Here, dω s the volume form of B E 0, 1. Ths s evaluated: r ρ 2 ρ n 1 dρ dω = ρ n+1 dρ B E 0,r Therefore n concluson, Theorem Gven a ont, we have 0 = rn+2 n + 2 V B E0, 1 B E 0,1 = rn+2 n + 2 n r n V B E0, r. V B, r V B E 0, r = 1 S 6n + 2 r2 + Or 4. Therefore we see that scalar curvature at a ont descrbes how the volume of geodesc ball centered at that ont devates from the volume of standard eucldean ball wth the same radus. Acknowledgments. It s a leasure to thank my mentor, Red Harrs, for hs atence n answerng my questons and revsng my aer. Everytme I sought for geometrc ntutons of concets such as Remann Curvature and Rcc curvature, he reared examles and dagrams to exlan to me. Red also rovded mortant advce of organzng a clear and beautful aer. Wthout hs dedcaton I would not have fnshed ths aer. References [1] Davd T. Guarrera, Nles G. Johnson and Homer F. Wolfe. The Taylor Exanson of a Remannan Metrc. htts:// n2/wolfe/rmn Metrc.df. dω.