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The Sasaki metric gives a natural way to equip $TM$ with a Riemannian metric in case $M$ is already equipped with a Riemanian metric. Question: Let $M$ be manifold equipped with a connection, is there a known natural way to equip $TM$ with a connection ?

Arctic Char
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Amr
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  • Have you tried finding the connection associated to the Sasaki metric and relating it to the connection from the Riemannian metric? – Aitor Iribar Lopez Jan 14 '22 at 17:13
  • I considered doing that but I still dont have good intuition for the Sasaki metric so perhaps it will be difficult to compute its levi civita connection. I thought to ask here first in case the construction is already known. If its not known, then I ll have to do what you suggested. – Amr Jan 14 '22 at 17:15
  • I also tried defining a natural connection on TM right away but I am not sure if my construction works – Amr Jan 14 '22 at 17:16
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    What precisely do you mean by a manifold equipped with a connection? I only know about connections on vector bundles, principal $G$-bundles, etc. – Ted Shifrin Jan 14 '22 at 17:56
  • Given $(M,\nabla)$, I'm not sure about a connection on the manifold $TM$, but the projection $\pi\colon TM\to M$ induces a connection $\pi^\nabla$ on the bundle $\pi^(TM) \to TM$. – Ivo Terek Jan 14 '22 at 22:36
  • @Ted "manifold with connection" is a shorthand for "manifold with a connection on its tangent bundle (as a vector bundle)". Affine differential geometry is a big thing (the first part of Postnikov's RG book deals mainly with this, there's a book by Nomizu and Sasaki, a chapter by Udo Simon on a volume of the handbook of differential geometry, etc.). It basically explores things that can happen in the particular case $E=TM$ (such as having torsion) – Ivo Terek Jan 14 '22 at 22:40
  • @Ivo That would have been my usual understanding, but the nature of this question was vague. There are, in particular, various connections, like projective connections. I assume this is just an affine connection, but who knows. – Ted Shifrin Jan 14 '22 at 22:48
  • Oh, I see now how it could have been more clear. I immediately understood it as a connection on the manifold $TM$. – Ivo Terek Jan 14 '22 at 22:53
  • Yes it is an affine connection on the manifold TM – Amr Jan 16 '22 at 14:48

2 Answers2

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The construction of the Sasaki metric relies on the fact that an affine connection $\nabla$ defines a splitting of $TTM$ into vertical and horizontal subbundles $TTM\cong VTM\oplus HTM$: A vector $v\in TTM$ is vertical if it is tangent to a fiber, and horizontal if it is the derivative of a parallel vector field along a curve. Both of these subbundles are canonically isomorphic to the pullback bundle $\pi_{TM}^*(TM)$: The pullback maps are given by the differential $d\pi_{TM}|_{HTM}$ for $HTM$ and (fiberwise) by the canonical isomorphism of vector spaces $T_vV\cong V$ for $VTM$.

The Sasaki metric uses this "sum of pullbacks" structure to induce a metric on $TTM$, but we can do the same thing with an affine connection, since there is a canonical affine connection induced on pullback bundles as well as Whitney sums.

Kajelad
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  • Aha, I guessed this construction before asking the question it is good that your answer reassures me I wasnt guessing in the wrong direction. Thanks – Amr Jan 17 '22 at 17:27
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$TM$ is a smooth manifold. Hence admits a Riemannian structure. Thus you may equip it with the levi civita connection.

Moe Khaled Bin-Lateef
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    Thats not a natural way to do it... – Amr Jan 14 '22 at 16:51
  • @Amr I understand what you mean, but don't you think that "natural way" is a bit vague? I voted to close your question. You should at least give some explanation what you expect to be the relation between connections on $M$ and on $TM$. – Paul Frost Jan 14 '22 at 23:38
  • To the downvoters: It is easy to downvote, but do you think the question is sufficiently clear? – Paul Frost Jan 14 '22 at 23:40
  • @PaulFrost Ofcourse, it is a vague question but I think this is okay. Ideally, mathematical activity should be about asking precisely defined questions and answering them by a rigorous proof in a suitable formal system. However, that's not the full story. There exists the pre-rigour stage which allows heurisitc arguments, experimentation, intuition, asking ill defined questions...etc. – Amr Jan 16 '22 at 14:36
  • @PaulFrost Perhaps, category theorists can make my naturality condition precise. I just assumed that all people would interpret the word naturally in a reasonable way, but I admit I don't have a precise defintion for what counts as natural. – Amr Jan 16 '22 at 14:42
  • @Amr then why isn’t the answer above a reasonable interpretation if you don’t have a precise definition? – Moe Khaled Bin-Lateef Jan 18 '22 at 17:16
  • @Mohammad While it is true that I don't have a precise definition of what natural is but I have a good approximate idea about what it is that allows me to reject your answer. The connection you equip $TM$ with has nothing to do with the connection on $M$, also your construction is not unique as it will depend on the underlying Riemannian metric or the underlying partition of unity chosen out of uncountably many possibilities – Amr Jan 19 '22 at 00:12
  • Suppose I had asked instead given a finite abelian group $G$, is there a way natural way to equip $Hom(G,G)$ with a group operation. One natural interpretation would be to equip it with the group structure resulting from pointwise addition. An unreasonable interpreation would be: Biject $Hom(G,G)$ with the cyclic group $Z_n$ for some $n$, then pull back the cyclic group structure to $Hom(G,G)$ – Amr Jan 19 '22 at 00:16
  • Sure, but in this setting, being "reasonable'" depends on the problem at hand. – Moe Khaled Bin-Lateef Jan 20 '22 at 18:21
  • Well, even though my question is not 100% accurate and is prone to trivializing useless answers like yours, I still don't regret asking it. Someone else was able to interpret my question reasonably and gave me a useful answer ;) Thanks for your attempt to help anyway :)) – Amr Feb 15 '22 at 20:58
  • @Amr why is it useless? It fostered a discussion, so I don't see why it would be useless. The answers taught me something and that means it has the potential to teach someone else something too... Isn't that the point of this site? To foster thought and discussion? hence I don't see why it would be useless. Furthermore, it's great you don't regret asking it. Why should you? – Moe Khaled Bin-Lateef Feb 15 '22 at 22:17