285G, Lecture 1: Flows on Riemannian manifolds – Exercises

Exercise 1. Show that

\dot{\hbox{Riem}}_{\alpha \beta \delta}^\gamma = \frac{1}{2} g^{\gamma \sigma} ( \nabla_\alpha \nabla_\delta \dot g_{\beta \sigma} - \nabla_\alpha \nabla_\sigma \dot g_{\delta \beta} - \nabla_\beta \nabla_\delta \dot g_{\alpha \sigma} + \nabla_\beta \nabla_\sigma \dot g_{\delta \alpha} (12)

is consistent with the antisymmetry properties of the Riemann tensor, and with the Bianchi identities, as presented in the previous lecture.

Exercise 2. Verify the derivation of

\dot{\hbox{Ric}}_{\alpha \beta} = - \frac{1}{2}\Delta_L \dot g_{\alpha \beta} - \frac{1}{2} \nabla_\alpha \nabla_\beta \hbox{tr}(\dot g) - \frac{1}{2} \nabla_\alpha \nabla^\gamma \dot g_{\beta \gamma} - \frac{1}{2} \nabla_\beta \nabla^\gamma \dot g_{\alpha \gamma} (13)

and

\dot R = - \hbox{Ric}^{\alpha \beta} \dot g_{\alpha \beta} - \Delta \hbox{tr}(\dot g) + \nabla^\alpha \nabla^\beta \dot g_{\alpha \beta} (15).

[Aside: – I wonder if there are more direct derivations of (13) and (15) that do not require one to go through so many computations. One can use (22) and (26) below as consistency checks for these formulae, but this does not quite seem sufficient.]

Exercise 3. If \gamma varies smoothly in time (but with static endpoints \gamma(a),\gamma(b), show that

\frac{d}{dt} L(\gamma) = \frac{1}{2} \int_\gamma \dot g( S, S )\ ds - \int_\gamma g( \nabla_S S, V )\ ds (17)

where at every point x=\gamma(u) of the curve, S=\gamma'(u)/g(\gamma'(u),\gamma'(u))^{1/2} is the unit tangent, and V=\dot\gamma(u) is the variation field. (Strictly speaking, one needs to work on the pullback tensor bundles on {}[a,b] rather than M in order to make the formula in (17) well defined.)

Exercise 4. Establish the first variation formula

\frac{d}{dt} d(x,y) = \inf g( X(y), S(y) ) - g( X(x), S(x) ),

where the infimum ranges over all minimal geodesics from x to y (which in particular determine the unit tangent vector S at x and at y).

Exercise 5. Verify

\nabla_\alpha X^\beta = \overline{\nabla}_\alpha X^\beta +  \Gamma_{\alpha\gamma}^\beta X^\gamma (37)

and

\Gamma_{\alpha \gamma}^\beta = \frac{1}{2} g^{\beta \delta} ( \overline{\nabla}_\alpha g_{\gamma \delta} + \overline{\nabla}_\gamma g_{\alpha \delta} - \overline{\nabla}_\delta g_{\alpha \gamma} ) (38).

Then use these formulae to give an alternate derivation of (5) and (8).

Exercise 6. Show that the expression

\overline{\hbox{Ric}}_{\alpha \beta} - \frac{1}{2} g^{\gamma \delta} \nabla_\gamma \nabla_\delta g_{\alpha \beta}  + \frac{1}{2} {\mathcal L}_X g_{\alpha \beta} + {\mathcal O}( g^{-2} \overline{\nabla} g \overline{\nabla} g ) (41)

for the Ricci curvature can be used to imply (13). Conversely, use (13) to recover (41) without performing an excessive amount of explicit computation. (Hint: first show that the Ricci tensor can be crudely expressed as \overline{\hbox{Ric}} + {\mathcal O}(g^{-1} \overline{\nabla}^2 g ) + {\mathcal O}(g^{-2} \overline{\nabla} g \overline{\nabla} g ). )

4 Responses to 285G, Lecture 1: Flows on Riemannian manifolds – Exercises

  1. Dexter says:

    Ene Tesseract ajild orohguim bhdaa. Asuuh zuil zunduu garaad bh yum. Python helnii talaar yumnuud bdag bolow uu?

  2. ttlc says:

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  3. t8m8r says:

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  4. gamgaa says:

    Sain uu. Asuuh yum baina. Shugaman Algebr dunguj uzej ehelj baigaa bolohoor zarim neg tun engiin baij boloh yumaa sain oilgodoggui ee.

    thm: U, V ni tus bur n bolon m zergiin vector ogtorgui bolog. edgeer ogtorgui tus buriin base-g {u_1, u_2, …,u_n}, {v_1, v_2,…,v_m} gej todorhoilj orhiy. ene uyed linear mapping(mongoloor yu gedeg yum boldoo) f:U->V -nii huvid f(u_1, u_2, …,u_n)=(v_1, v_2,…,v_m)A geh nuhtsuliig hangah (m,n) hemjeest A matrix hargalzana.
    Mun esregeeree (m,n) hemjeest A matrix-n huvid linear mapping f:U->V hargalzana.

    A-g f-n matrix representation gene.

    za deerh theorem-g bodlogond sain ashiglaj chaddaggui ee.
    jishee ni
    base {e_1, e_2, e_3, e_4} -d hargalzan (4,4) hemjeestei neg matrix todorhoilogdood ugugdchihsun baina. ene uyed {e_1+e_2, e_2+e_3, e_2-e_4, e_2+e_4} gesen base-d hargalzan matrix representation-g yaj oloh ve? (end e_k gedeg ni k dehi component ni 1 busad ni 0 baih vector)

    uul ni hamgiin l engiin zuil baih. daanch theoremoo nariin uhaj oilgoj chadahgui baina uu. neg l burheg baina.

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